#  Tightening geometric and dynamical constraints on dark energy and gravity:
galaxy clustering, intrinsic alignment and kinetic Sunyaev-Zel’dovich effect

Teppei Okumura<EMAIL_ADDRESS>Academia Sinica Institute of
Astronomy and Astrophysics (ASIAA), No. 1, Section 4, Roosevelt Road, Taipei
10617, Taiwan  Kavli Institute for the Physics and Mathematics of the Universe
(WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan  Atsushi
Taruya  Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto
606-8502, Japan  Kavli Institute for the Physics and Mathematics of the
Universe (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

###### Abstract

Conventionally, in galaxy surveys, cosmological constraints on the growth and
expansion history of the universe have been obtained from the measurements of
redshift-space distortions and baryon acoustic oscillations embedded in the
large-scale galaxy density field. In this paper, we study how well one can
improve the cosmological constraints from the combination of the galaxy
density field with velocity and tidal fields, which are observed via the
kinetic Sunyaev-Zel’dovich (kSZ) and galaxy intrinsic alignment (IA) effects,
respectively. For illustration, we consider the deep galaxy survey by Subaru
Prime Focus Spectrograph, whose survey footprint perfectly overlaps with the
imaging survey of the Hyper Suprime-Cam and the CMB-S4 experiment. We find
that adding the kSZ and IA effects significantly improves cosmological
constraints, particularly when we adopt the non-flat cold dark matter model
which allows both time variation of the dark energy equation-of-state and
deviation of the gravity law from general relativity. Under this model, we
achieve $31\%$ improvement for the growth index $\gamma$ and $>35\%$
improvement for other parameters except for the curvature parameter, compared
to the case of the conventional galaxy-clustering-only analysis. As another
example, we also consider the wide galaxy survey by the Euclid satellite, in
which shapes of galaxies are noisier but the survey volume is much larger. We
demonstrate that when the above model is adopted, the clustering analysis
combined with kSZ and IA from the deep survey can achieve tighter cosmological
constraints than the clustering-only analysis from the wide survey.

##  I Introduction

The baryon acoustic oscillations (BAO) Peebles and Yu (1970); Sunyaev and
Zeldovich (1970); Eisenstein and Hu (1998) and redshift-space distortions
(RSD) Peebles (1980); Kaiser (1987); Hamilton (1998) imprinted in large-scale
galaxy distribution have been widely used as powerful tools to constrain the
expansion and growth history of the Universe. Measurements of these signals
enable galaxy clustering from redshift surveys to be one of the most promising
probes to clarify the origin of the late-time cosmic acceleration, which could
be explained by dark energy or modification of gravity Peacock et al. (2001);
Tegmark et al. (2004); Eisenstein et al. (2005); Okumura et al. (2008); Guzzo
et al. (2008); Weinberg et al. (2013); Aubourg et al. (2015); Okumura et al.
(2016); Alam et al. (2017); Suyu et al. (2018); Alam et al. (2021). Upcoming
spectroscopic galaxy surveys, including the Subaru Prime Focus Spectrograph
(PFS) Takada et al. (2014), the Dark Energy Spectroscopic Instrument (DESI)
DESI Collaboration et al. (2016), the Euclid space telescope Laureijs et al.
(2011); Capak et al. (2019); Euclid Collaboration et al. (2020), and the Nancy
Grace Roman Space Telescope Spergel et al. (2013a, b); Eifler et al. (2021),
aim to constrain the dark energy equation-of-state and deviation of the
gravitational law from general relativity (GR) with a precision at the sub-
percent level.

In order to maximize the information encoded in the galaxy distribution in the
large-scale structure (LSS) and to constrain cosmological parameters as
tightly as possible, one needs to effectively utilize synergies between galaxy
redshift surveys and other observations. In this respect, there is a growing
interest of using two effects below as new probes of the LSS to improve
cosmological constraints, complementary to the conventional galaxy clustering
analysis. The first is the kinetic Sunyaev-Zel’dovich (kSZ) effect Sunyaev and
Zeldovich (1980); Ostriker and Vishniac (1986), which can be observed via the
measurement of cluster velocities by a synergy between galaxy surveys and
cosmic microwave background (CMB) experiments. Theoretical and forecast
studies suggest that kSZ measurements could provide robust tests of dark
energy and modified gravity theories on large scales Hernández-Monteagudo et
al. (2006); Okumura et al. (2014); Mueller et al. (2015); Sugiyama et al.
(2016, 2017); Zheng (2020). The kSZ effect has been detected through the
cross-correlations of CMB data with galaxy positions from various redshift
surveys Hand et al. (2012); De Bernardis et al. (2017); Sugiyama et al.
(2018); Planck Collaboration et al. (2018); Li et al. (2018); Calafut et al.
(2021); Chaves-Montero et al. (2021); Chen et al. (2022).

The second probe is intrinsic alignment (IA) of galaxy shapes with the
surrounding large-scale matter density field. The IA was originally proposed
as a source of systematic effects on the measurement of the cosmological
gravitational lensing Croft and Metzler (2000); Heavens et al. (2000); Lee and
Pen (2000); Pen et al. (2000); Catelan et al. (2001); Jing (2002); Hirata and
Seljak (2004); Heymans et al. (2004); Mandelbaum et al. (2006); Hirata et al.
(2007); Okumura et al. (2009); Okumura and Jing (2009); Blazek et al. (2011);
Singh et al. (2015). However, since the spatial correlation of IA follows the
gravitational tidal field induced by the LSS, it contains valuable information
and is considered as a cosmological probe complimentary to the galaxy
clustering Schmidt and Jeong (2012); Faltenbacher et al. (2012); Chisari and
Dvorkin (2013); Chisari et al. (2016); Okumura and Taruya (2020); Okumura et
al. (2020); Taruya and Okumura (2020); Masaki et al. (2020); Akitsu et al.
(2021a, b). Ongoing and future galaxy surveys focus on observing LSS at higher
redshifts, $z>1$, at which the emission line galaxies (ELG) would be an ideal
tracer of the LSS Laureijs et al. (2011); Takada et al. (2014); Tonegawa et
al. (2015); Okada et al. (2016); Dawson et al. (2016); DESI Collaboration et
al. (2016). Although IA has not yet been detected for ELG Mandelbaum et al.
(2006, 2011); Tonegawa et al. (2018); Tonegawa and Okumura (2022), recent work
Shi et al. (2021) has proposed an effective estimator to determine the IA of
dark-matter halos using ELG, enhancing the signal-to-noise ratio at a
statistically significant level. In any case, the accurate determination of
galaxy shapes is of critical importance for IA to be a powerful tool to
constrain cosmology. Thus, the synergy between imaging and spectroscopic
surveys is essential because the accurate galaxy shapes and positions are
determined from the former and latter, respectively.

In this paper, using the Fisher matrix formalism, we simultaneously analyze
the velocity and tidal fields observed by the kSZ and IA effects,
respectively, together with galaxy clustering. The combination of galaxy
clustering with either IA or kSZ has been studied in earlier studies (e.g.,
Mueller et al., 2015; Sugiyama et al., 2017; Taruya and Okumura, 2020). This
is the first joint analysis of these three probes and we want to see if
cosmological constraints can be further improved by combining the combination.
We emphasize that the question we want to address is not trivial at all
because these probes utilize the information embedded in the same underlying
matter fluctuations. Nevertheless, a key point is that these different probes
suffer from different systematic effects, and can be in practice complementary
to each other, thus used as a test for fundamental observational issues, such
as the Hubble tension Verde et al. (2019), if the constraining power of each
probe is similar. Furthermore, analyzing the kSZ and IA simultaneously enables
us to study the correlation of galaxy orientations in phase space as proposed
in our recent series of work Okumura et al. (2017, 2018, 2019); Okumura and
Taruya (2020); Okumura et al. (2020). For our forecast, we mainly consider the
PFS-like deep galaxy survey Takada et al. (2014) which overlaps with the
imaging survey of the Hyper Suprime-Cam (HSC) Miyazaki et al. (2012); Aihara
et al. (2018) and the CMB Stage-4 experiment (CMB-S4) Abazajian et al. (2016).
To see how the cosmological gain by adding the IA and kSZ effects to galaxy
clustering can be different for different survey geometries, we also analyze
the Euclid-like wide galaxy survey Laureijs et al. (2011); Euclid
Collaboration et al. (2020).

The rest of this paper is organized as follows. In section II we briefly
summarize geometric and dynamic quantities to be constrained. Section III
presents power spectra of galaxy density, velocity and ellipticity fields and
their covariance matrix. We perform a Fisher matrix analysis and present
forecast constraints in section IV, with some details further discussed in
section V. Our conclusions are given in section VI. Appendix A describes the
CMB prior used in this paper. In appendix B, we present conservative forecast
constraints by restricting the analysis to large scales where linear
perturbation theory is safely applied.

##  II Preliminaries

###  II.1 Distances

The comoving distance to a galaxy at redshift $z$, $\chi(z)$, is given by

$\chi(z)=\int^{z}_{0}\frac{cdz^{\prime}}{H(z^{\prime})}\,,$ (1)

with $c$ being the speed of light. The function $H(z)$ is the Hubble parameter
which describes the expansion rate of the universe. Writing it as
$H(z)=H_{0}E(z)$, we define the present-day value of the Hubble parameter by
$H_{0}\equiv H(z=0)$, which is often characterized by the dimensionless Hubble
constant, $h$, as $H_{0}=100\,h\,{\rm km~{}s}^{-1}{\rm Mpc}^{-1}$. Then the
time-dependent function $E(z)$ is obtained from the Friedmann equation, and is
expressed in terms of the (dimensionless) density parameters. In this paper,
we consider the universe whose cosmic expansion is close to that in the
standard cosmological model, with the dark energy having the time-varying
equation of state. Allowing also the non-flat geometry, the function $E(z)$ is
given by

$\displaystyle E^{2}(z)$ $\displaystyle=$ $\displaystyle\Omega_{\rm
m}(1+z)^{3}+\Omega_{K}(1+z)^{2}$ (2) $\displaystyle+$
$\displaystyle\Omega_{\rm
DE}(1+z)^{3(1+w_{0}+w_{a})}\exp{\left[-3w_{a}\frac{z}{1+z}\right]},$

where $\Omega_{\rm m}$, $\Omega_{\rm DE}$ and $\Omega_{K}$ are the present-day
energy density fractions of matter, dark energy and curvature, respectively,
with $\Omega_{\rm m}+\Omega_{\rm DE}+\Omega_{K}=1$. In equation (2), the time-
varying equation-of-state parameter for dark energy, denoted by $w(z)$, is
assumed to be described by a commonly used and well tested parameterization
Chevallier and Polarski (2001); Linder (2003):

$w(z)=w_{0}+w_{a}\frac{z}{1+z}=w_{0}+w_{a}(1-a),$ (3)

where $a=(1+z)^{-1}$ is the scale factor, and $w_{0}$ and $w_{a}$ characterize
the constant part and the amplitude of time variation of the dark energy
equation of state, respectively (see e.g., Ref. Colgáin et al. (2021), which
studied how the different parameterization of $w(z)$ affects the constraining
power of the deviation of a cosmological constant.)

The angular diameter distance, $D_{\rm A}(z)$, is given as

$D_{\rm
A}(z)=(1+z)^{-1}\frac{c}{H_{0}}S_{K}\left(\frac{\chi(z)}{c/H_{0}}\right),$ (4)

where

$\displaystyle
S_{K}(x)=\left\\{\begin{array}[]{lll}\sin{\left(\sqrt{-\Omega_{K}}x\right)}/\sqrt{-\Omega_{K}}&&\Omega_{K}<0,\\\
x&&\Omega_{K}=0,\\\
\sinh{\left(\sqrt{\Omega_{K}}x\right)}/\sqrt{\Omega_{K}}&&\Omega_{K}>0.\\\
\end{array}\right.$ (8)

Negative and positive values of $\Omega_{K}$ correspond to the closed and open
universe, respectively. The geometric quantities, $D_{\rm A}(z)$ and $H(z)$,
are the key quantities we directly constrain from the measurement of the BAO
imprinted in the power spectra.

###  II.2 Perturbations

Density perturbations for a given component $i$ ($i=\\{\rm m,g\\}$ for matter
and galaxies, respectively) are defined by the density contrast from the mean
$\bar{\rho}_{i}(z)$,

$\delta_{i}(\bm{x};z)\equiv\rho_{i}(\bm{x};z)/\bar{\rho}_{i}(z)-1.$ (9)

Throughout the paper, we assume the linear relation for the galaxy bias with
which the galaxy density fluctuation $\delta_{g}$ is related to the matter
fluctuation $\delta_{m}$ through $\delta_{g}=b_{g}\,\delta_{m}$ Kaiser (1984).
Then, an important quantity to characterize the evolution of the density
perturbation is the growth rate parameter, defined as

$f(z)=-\frac{d\ln D(z)}{d\ln(1+z)}=\frac{d\ln D(a)}{d\ln a},$ (10)

where $D(z)$ is the linear growth factor of the matter perturbation,
$D(z)=\delta_{m}(\bm{x};z)/\delta_{m}(\bm{x};0)$. The parameter $f$ quantifies
the cosmological velocity field and the speed of structure growth, and thus is
useful for testing a possible deviation of the gravity law from GR Guzzo et
al. (2008). For this purpose, it is common to parameterize the $f$ parameter
as

$f(z)=\left[\Omega_{\rm m}(z)\right]^{\gamma},$ (11)

where $\Omega_{\rm m}(z)=\Omega_{\rm m}(1+z)^{3}/E^{2}(z)$ is the time-
dependent matter density parameter and the index $\gamma$ specifies a model of
gravity, e.g., $\gamma\approx 6/11$ for the case of GR Peebles (1980); Linder
(2005).

It is known that a class of modified gravity models exactly follows the same
background evolution as in the $\Lambda$CDM model. However, the evolution of
density perturbations can be different in general (see, e.g., Ref. Matsumoto
et al. (2020) for degeneracies between the expansion and growth rates for
various gravity models). Thus, it is crucial to simultaneously constrain the
expansion and growth rate of the universe to distinguish between modified
gravity models.

##  III Power spectra and the Fisher matrix

In this paper, we consider three cosmological probes observed in redshift
space, i.e., density, velocity and ellipticity (tidal) fields. While
nonlinearity of the density field has been extensively studied and a precision
modeling of its redshift-space power spectrum has been developed (e.g.,
Peacock and Dodds, 1994; Scoccimarro, 2004; Taruya et al., 2010; Okumura et
al., 2015), the understanding of the nonlinearities of velocity and tidal
fields are relatively poor. However, there are several numerical and
theoretical studies discussed beyond the linear theory, among which a
systematic perturbative treatment has been also exploited (See, e.g., Refs.
Okumura et al. (2014); Sugiyama et al. (2016, 2017) and Blazek et al. (2015,
2019); Vlah et al. (2020) for the nonlinear statistics of velocity and tidal
fields, respectively). It is thus expected that a reliable theoretical
template of their power spectra would be soon available, and an accessible
range of their templates can reach, at least, at the weakly nonlinear regime.
Hence, in our analysis, we consider the weakly nonlinear scales of $k\leq
0.2\,h\,{\rm Mpc}^{-1}$, as our default setup. Nevertheless, in order for a
robust and conservative cosmological analysis, we do not use the shape
information of the underlying matter power spectrum, which contains ample
cosmological information but is more severely affected by the nonlinearities.
That is, our focus in this paper is the measurements of BAO scales and RSD
imprinted in the power spectra, and through the geometric and dynamical
constraints on $D_{\rm A}(z)$, $H(z)$ and $f(z)$, we further consider
cosmological constraints on models beyond the $\Lambda$ cold dark matter
($\Lambda$CDM) model. In appendix B, we perform a more conservative forecast
by restricting th analysis to large scales, $k\leq 0.1\,h\,{\rm Mpc}^{-1}$,
where linear perturbation theory predictions can be safely applied.

In what follows, we discuss how well one can maximize the cosmological
information obtained from the BAO and RSD measurements, based on the linear
theory predictions. While the linear-theory based template is no longer
adequate at weakly nonlinear scales, the signal and information contained in
the power spectrum can be in general maximized as long as we consider the
Gaussian initial condition. In this respect, the results of our analysis
presented below may be regarded as a theoretical upper bound on the
cosmological information one can get. Furthermore, we assume a plane-parallel
approximation for the cosmological probes Szalay et al. (1998); Szapudi
(2004), taking the $z$-axis to be the line-of-sight direction. While properly
taking into account the wide-angle effect provides additional cosmological
constraints (see, e.g., Refs. Taruya et al. (2020), Castorina and White
(2020); Shiraishi et al. (2020); Shiraishi et al. (2021a) and Shiraishi et al.
(2021b) for the studies of the wide-angle effects on density, velocity and
ellipticity fields, respectively), we leave the inclusion of this effect to
our analysis as future work.

###  III.1 Density, velocity and ellipticity fields

In this subsection, based on the linear theory description, we write down the
explicit relation between cosmological probes observed in redshift space to
the matter density field. First, the density field of galaxies in redshift
space, which we denote by $\delta_{g}^{S}$, is a direct observable in galaxy
redshift surveys, and in Fourier space, it is related to the underlying
density field of matter in real space on large scales, through
$\delta_{g}^{S}(\bm{k};z)=K_{g}(\mu;z)\delta_{m}(\bm{k};z)$. The factor
$K_{g}$ is the so-called linear Kaiser factor given by Kaiser (1987); Hamilton
(1992); Okumura and Jing (2011),

$K_{g}(\mu;z)=b_{g}(z)+f(z)\mu^{2},$ (12)

where $b_{g}$ is the galaxy bias and $\mu$ is the directional cosine between
the wavevector and line-of-sight direction,
$\mu=\hat{\bm{k}}\cdot\hat{\bm{z}}$, with a hat denoting a unit vector. Note
that setting $f$ to zero, the above equation is reduced to the Fourier
counterpart of $\delta_{g}$ in equation (9).

Next, the cosmic velocity field is related to the density field through the
continuity equation Fisher (1995); Strauss and Willick (1995). The observable
through the kSZ effect is the line-of-sight component of the velocity,
$v_{\parallel}$, and in linear theory, we have (in Fourier space)
$v_{\parallel}(\bm{k};z)=if(z)\mu aH\delta_{m}(\bm{k};z)/k$. To be precise,
the kSZ effect measures the temperature distortion of CMB, $\delta T$,
detected at the position of foreground galaxies. It is explicitly written in
Fourier space as $\delta
T(\bm{k};z)=(T_{0}\tau/c)v_{\parallel}(\bm{k};z)=iK_{v}(\bm{k};z)\delta_{m}(k;z)$,
where

$K_{v}(k,\mu;z)=\frac{T_{0}\tau}{c}\frac{f(z)\mu aH(z)}{k}~{},$ (13)

with $\tau$ being the optical depth. Since the distance to tracers of the
velocity field is measured by redshift, the observed velocity field is
affected by RSD, similarly to the density field in redshift space. Unlike the
density field, however, the RSD contribution to the redshift-space velocity
field appears at higher order Okumura et al. (2014). Thus, at leading order,
the velocity field traced in redshift space coincides with that in real space
in linearized theory, $v_{\parallel}^{S}=v_{\parallel}$. Note that the kSZ
effect, which appears as secondary CMB anisotropies, is given by a line-of-
sight integral of the velocity field, and thus the expression of Eq. (13) is
just an approximation. We discuss the validity of this approximation in
section V.3.

An alternative way to measure the velocity field $v_{\parallel}$ without
observing the temperature distortion is to use velocity surveys, which enable
us to uniquely constrain $f(z)$ Strauss and Willick (1995). We, however, do
not consider observables from peculiar velocity surveys. The main reason is
that these observations are limited to the nearby universe ($z\approx 0$)
while we consider joint constraints with other probes from a single
observation of the LSS. Thus, throughout this paper we refer the velocity
field as the temperature distortion $\delta T$.

Table 1: Statistics and their abbreviations considered for given probes. Note that when two fields, $A$ and $B$, are considered, we use not only the auto-correlations ($P_{AA}$ and $P_{BB}$) but also the cross correlation, $P_{AB}$.  Probes | Statistics | Abbreviations | No. of | Parameters $\\{\theta_{\alpha}\\}$
---|---|---|---|---
parameters $N_{\theta}$ | nuisance | geometric/dynamical
Clustering | $P_{gg}$ | $g$ | 4 | $b\sigma_{8}$ | $f\sigma_{8},H,D_{\rm A}$
kSZ | $P_{vv}$ | $v$ | 4 | $\tau$ | $f\sigma_{8},H,D_{\rm A}$
IA | $P_{EE}$ | $E$ | 3 | $A_{\rm IA}$ | $H,D_{\rm A}$
Clustering+IA | $P_{gg}+P_{EE}+P_{gE}$ | $g+E$ | 5 | $b\sigma_{8},A_{\rm IA}$ | $f\sigma_{8},H,D_{\rm A}$
Clustering+kSZ | $P_{gg}+P_{vv}+P_{gv}$ | $g+v$ | 5 | $b\sigma_{8},\tau$ | $f\sigma_{8},H,D_{\rm A}$
IA+kSZ | $P_{EE}+P_{vv}+P_{vE}$ | $v+E$ | 5 | $A_{\rm IA},\tau$ | $f\sigma_{8},H,D_{\rm A}$
Clustering+IA+kSZ | $P_{gg}+P_{EE}+P_{vv}+P_{gE}+P_{gv}+P_{vE}$ | $g+v+E$ | 6 | $b\sigma_{8},A_{\rm IA},\tau$ | $f\sigma_{8},H,D_{\rm A}$

Finally, we use ellipticities of galaxies as a tracer of the tidal field. The
two-component ellipticity of galaxies is defined as

$\gamma_{(+,\times)}(\bm{x};z)=\frac{1-q}{1+q}\left(\cos{(2\phi_{x})},\sin{(2\phi_{x})}\right),$
(14)

where $\phi_{x}$ is the position angle of the major axis relative to the
reference axis, defined on the plane normal to the line-of-sight direction,
and $q$ is the minor-to-major axis ratio of a galaxy shape. We set $q$ to zero
for simplicity Okumura et al. (2009). As a tracer of LSS, a leading-order
description of the ellipticity field is to relate $\gamma_{(+,\times)}$
linearly to the tidal gravitational field, known as the linear alignment (LA)
model Catelan et al. (2001); Hirata and Seljak (2004); Okumura et al. (2019);
Okumura and Taruya (2020). In Fourier space, this is given by

$\gamma_{(+,\times)}(\bm{k};z)=b_{K}(z)\left(k_{x}^{2}-k_{y}^{2},2k_{x}k_{y}\right)\frac{\delta_{m}(\bm{k};z)}{k^{2}}.$
(15)

Just like the velocity field, the ellipticity field is not affected by RSD in
linear theory Okumura and Taruya (2020). We then define E-/B-modes,
$\gamma_{(E,B)}$, which are the rotation-invariant decomposition of the
ellipticity field Crittenden et al. (2002),

$\displaystyle\gamma_{E}(\bm{k};z)+i\gamma_{B}(\bm{k};z)=e^{-2i\phi_{k}}\left\\{\gamma_{+}(\bm{k};z)+i\gamma_{\times}(\bm{k};z)\right\\},$
(16)

where $\phi_{k}$ is the azimuthal angle of the wavevector projected on the
celestial sphere (Note that $\phi_{k}$ has nothing to do with the directional
cosine of the wavevector, and thus $\phi_{k}\neq\cos^{-1}{\mu}$). By writing
$\gamma_{(E,B)}(\bm{k};z)=K_{(E,B)}(\mu;z)\delta_{m}(\bm{k};z)$, we have
$K_{B}=0$ and

$K_{E}(\mu;z)=b_{K}(z)(1-\mu^{2}).$ (17)

In Eq. (15) or (16), the parameter $b_{K}$ quantifies the response of
individual galaxy shapes to the tidal field of LSS, and it is conventionally
characterized by introducing the parameter $A_{\rm IA}$ as follows (e.g.,
Schmitz et al., 2018; Kurita et al., 2021):

$b_{K}(z)=0.01344A_{\rm IA}(z)\Omega_{\rm m}/D(z).$ (18)

Note that the parameter $A_{\rm IA}$ generally depends on properties of the
given galaxy population as well as redshift. The analysis of numerical
simulations, however, demonstrated that for fixed galaxy/halo properties,
$A_{\rm IA}$ is nearly redshift-independent Kurita et al. (2021). We thus
treat $A_{\rm IA}$ as a constant throughout this paper.

###  III.2 Linear power spectra of the three fields

As summarized in the previous subsection, the three cosmological fields, i.e.
density, velocity and ellipticity, are related to the matter field linearly
through the coefficients, $K_{g}$, $K_{v}$ and $K_{E}$, respectively. Provided
their explicit expressions, we can analytically compute the auto-power spectra
of these fields and their cross-power spectra. There are in total six power
spectra measured in redshift space, each of which exhibits anisotropies
characterized by the $\mu$ dependence Ballinger et al. (1996); Seo and
Eisenstein (2003); Matsubara (2004); Okumura and Taruya (2020). Writing these
spectra as $P_{ij}(\bm{k};z)=P_{ij}(k,\mu;z)$ with $i,j=\\{g,v,E\\}$, they are
expressed in a concise form as,

$\displaystyle P_{ij}(k,\mu;z)$
$\displaystyle=K_{i}(k,\mu;z)K_{j}(k,\mu;z)P_{\rm lin}(k;z),$ (19)

where $P_{\rm lin}(k;z)$ is the linear power spectrum of matter fluctuation in
real space. The normalization of the density fluctuation is characterized by
the $\sigma_{8}$ parameter, defined by the linear RMS density fluctuation
within a sphere of radius $8h^{-1}$Mpc, and thus $P_{\rm
lin}(k;z)\propto\sigma_{8}^{2}(z)$. While each of the three auto-power
spectra, $P_{gg}$, $P_{vv}$ and $P_{EE}$, can be measured from each of the
three individual probes, namely galaxy clustering, kSZ and IA, respectively,
the cross-power spectra become measurable only when two probes are
simultaneously made available.111Note that this terminology is different from
that used in past studies: while in this paper the kSZ and IA power spectra
stand for only $P_{vv}$ and $P_{EE}$, respectively, the past studies included
the cross-power spectrum with density field, $P_{gv}$ and $P_{gE}$, into kSZ
and IA spectra. Particularly, the correlation between velocity and ellipticity
fields, $P_{vE}$, has been proposed recently by our earlier studies and it can
be probed by the joint analysis of the kSZ (or peculiar velocities) and IA
effects Okumura et al. (2017, 2018, 2019); Okumura and Taruya (2020); Okumura
et al. (2020); van Gemeren and Chisari (2020). Table 1 summarizes all the
statistics used in this paper.

To measure the power spectra, the observed galaxy positions measured with
redshift and angular position need to be converted into the comoving positions
by introducing a reference cosmology, with a help of equations (1) and (4). An
apparent mismatch between the reference and true cosmology causes a geometric
distortion in the measured power spectra, which is yet another anisotropy
known as the Alcock-Paczynski (AP) effect Alcock and Paczynski (1979). This AP
effect has been extensively investigated for the galaxy power spectrum in
redshift space Ballinger et al. (1996); Matsubara and Suto (1996); Seo and
Eisenstein (2003); Blazek et al. (2014). The AP effect on the kSZ and IA
statistics has been studied relatively recently by Refs. Sugiyama et al.
(2017) and Taruya and Okumura (2020), respectively. In all of the six power
spectra, $P_{ij}$, their observable counterpart $P_{ij}^{\rm obs}$ are related
to the true ones through the relation,

$\displaystyle P_{ij}^{{\rm obs}}\left(k_{\perp}^{\rm fid},k_{\parallel}^{{\rm
fid}};z\right)=\frac{H(z)}{H^{\rm fid}(z)}\left\\{\frac{D_{\rm A}^{\rm
fid}(z)}{D_{\rm
A}(z)}\right\\}^{2}P_{ij}\left(k_{\perp},k_{\parallel};z\right),$ (20)

where $k_{\perp}$ and $k_{\parallel}$ are the wavenumber perpendicular and
parallel to the line of sight,
$(k_{\perp},k_{\parallel})=k(\sqrt{1-\mu^{2}},\mu)$. The quantities $D_{\rm
A}^{\rm fid}(z)$ and $H^{\rm fid}(z)$ are the angular diameter distance and
expansion rate computed from fiducial cosmological parameters in the reference
cosmology, and $k_{\parallel}^{\rm fid}=k_{\parallel}H^{\rm fid}(z)/H(z)$ and
$k_{\perp}^{\rm fid}=k_{\perp}\,D_{\rm A}(z)/D_{\rm A}^{\rm fid}(z)$. The
prefactor $\frac{H(z)}{H^{\rm fid}(z)}\left\\{\frac{D_{\rm A}^{\rm
fid}(z)}{D_{\rm A}(z)}\right\\}^{2}$ accounts for the difference in the cosmic
volume in different cosmologies.

As formulated above, $K_{g}$, $K_{v}$ and $K_{E}$ respectively contain two
$(b,f)$, two $(\tau,f)$, and one $(A_{\rm IA})$ parameters, and all the power
spectra depend on $(H,D_{\rm A})$ through the AP effect (see table 1). Thus,
we have six parameters in total, $\theta_{\alpha}=(b\sigma_{8},A_{\rm
IA}\sigma_{8},\tau,f\sigma_{8},H,D_{\rm A})$, among which the first three are
nuisance parameters that we want to marginalize over. The latter three
parameters carry the cosmological information which characterize the growth of
structure and geometric distances, and are determined by measuring the
anisotropies in the power spectra.

###  III.3 Covariance matrix

Figure 1: Flowchart of our Fisher matrix analysis from dynamical and geometric
constraints to cosmological parameter constraints. The Fisher matrices of the
LSS probes, namely galaxy clustering, IA and kSZ, are given for each redshift
bin $z_{k}$ at the upper left. The Fisher matrix from the CMB prior is given
at the lower left.

Writing all the power spectra obtained from the galaxy clustering, kSZ and IA
as
$P_{a}=(P_{gg},P_{EE},P_{vv},P_{gE},P_{gv},P_{vE})$, we will below examine
several forecast analysis with a different number of power spectra, which we
denote by $N_{P}$. Specifically, depending on how many probes are
simultaneously available, we consider seven possible cases with $N_{P}=1$, $3$
or $6$, summarized in table 1. Correspondingly, the covariance matrix ${\rm
Cov}_{ab}$ becomes a $N_{P}\times N_{P}$ matrix, defined as ${\rm
Cov}_{ab}(k,\mu;z)=\left\langle P_{a}P_{b}\right\rangle-\left\langle
P_{a}\right\rangle\left\langle P_{b}\right\rangle$, for a given wavevector,
${\bf k}=(k,\mu)$. The full $6\times 6$ Gaussian covariance matrix reads

$\displaystyle{\rm
Cov}_{ab}(k,\mu;z)=\left[\begin{array}[]{cccccc}2\\{\widetilde{P}_{gg}\\}^{2}&2\\{P_{gE}\\}^{2}&2\\{P_{gv}\\}^{2}&2\widetilde{P}_{gg}P_{gE}&2\widetilde{P}_{gg}P_{gv}&2P_{gv}P_{gE}\\\
2\\{P_{gE}\\}^{2}&2\\{\widetilde{P}_{EE}\\}^{2}&2\\{P_{vE}\\}^{2}&2P_{gE}\widetilde{P}_{EE}&2P_{gE}P_{vE}&2\widetilde{P}_{EE}P_{vE}\\\
2\\{P_{gv}\\}^{2}&2\\{P_{vE}\\}^{2}&2\\{\widetilde{P}_{vv}\\}^{2}&2P_{gv}P_{vE}&2\widetilde{P}_{vv}P_{gv}&2P_{vE}\widetilde{P}_{vv}\\\
2\widetilde{P}_{gg}P_{gE}&2P_{gE}\widetilde{P}_{EE}&2P_{gv}P_{vE}&\widetilde{P}_{gg}\widetilde{P}_{EE}+\\{P_{gE}\\}^{2}&\widetilde{P}_{gg}P_{vE}+P_{gE}P_{gv}&P_{gv}\widetilde{P}_{EE}+P_{gE}P_{vE}\\\
2\widetilde{P}_{gg}P_{gv}&2P_{gE}P_{vE}&2\widetilde{P}_{vv}P_{gv}&\widetilde{P}_{gg}P_{vE}+P_{gE}P_{gv}&\widetilde{P}_{gg}\widetilde{P}_{vv}+\\{P_{gv}\\}^{2}&P_{gE}\widetilde{P}_{vv}+P_{gv}P_{vE}\\\
2P_{gv}P_{gE}&2\widetilde{P}_{EE}P_{vE}&2P_{vE}\widetilde{P}_{vv}&P_{gv}\widetilde{P}_{EE}+P_{gE}P_{vE}&P_{gE}\widetilde{P}_{vv}+P_{gv}P_{vE}&\widetilde{P}_{EE}\widetilde{P}_{vv}+\\{P_{vE}\\}^{2}\\\
\end{array}\right]~{},$ (27)

where $\widetilde{P}_{ii}=\widetilde{P}_{ii}(k,\mu;z)$ denotes an auto-power
spectrum ($i=\\{g,v,E\\}$) including the shot noise. Assuming the Poisson shot
noise, we have

$\displaystyle\widetilde{P}_{gg}=P_{gg}+\frac{1}{n_{g}},$ (28)
$\displaystyle\widetilde{P}_{vv}=P_{vv}+\left(1+R_{N}^{2}\right)\left(\frac{T_{0}\tau}{c}\right)^{2}\frac{(faH\sigma_{d})^{2}}{n_{v}},$
(29)
$\displaystyle\widetilde{P}_{EE}=P_{EE}+\frac{\sigma_{\gamma}^{2}}{n_{\gamma}},$
(30)

where the quantities $n_{g}$, $n_{v}$ and $n_{\gamma}$ are the number density
of the galaxies obtained from galaxy clustering, kSZ and IA observations,
respectively. Though different notations are explicitly used for these three
samples, $n_{g}=n_{v}=n_{\gamma}$ when one considers a single galaxy
population for the analysis. When one uses a single galaxy population as a
tracer of the density, velocity and ellipticity fields, there should be a shot
noise contribution in the cross correlations. Such a noise term, however,
vanishes because $\langle v_{\parallel}\rangle=0$ Sugiyama et al. (2017) and
$\langle\gamma_{E}\rangle=0$ Taruya and Okumura (2020).

In the shot noise terms of $P_{vv}$ and $P_{EE}$, there appear factors
$\sigma_{d}=\sqrt{\langle v_{\parallel}^{2}\rangle}$ and
$\sigma_{\gamma}=\sqrt{\langle\gamma_{E}^{2}\rangle}$, which respectively
represent the velocity dispersion and shape noise of galaxies, respectively.
Using perturbation theory, $\sigma_{d}$ can be evaluated as

$\sigma_{d}^{2}=\frac{1}{3}\int\frac{d^{3}q}{(2\pi)^{3}}\frac{P_{\theta\theta}(q;z)}{q^{2}}=\frac{1}{6\pi^{2}}\int
dqP_{\theta\theta}(q;z),$ (31)

where $P_{\theta\theta}$ is the power spectrum of velocity divergence. In the
limit of linear theory, we have $P_{\theta\theta}=P_{{\rm lin}}$, and in the
standard cosmological model, it is predicted to give $aH\sigma_{d,{\rm
lin}}\simeq 600D(z)~{}{\rm km}/{\rm s}$ Vlah et al. (2012), and hence
$faH\sigma_{d,{\rm lin}}\simeq 600f(z)D(z)\approx 300~{}{\rm km}/{\rm s}$ over
the redshift considered in this work. Finally, the parameter $R_{N}$ is the
inverse signal-to-noise ratio of the kSZ temperature fluctuations Sugiyama et
al. (2017). The rms noise for the kSZ measurement of the CMB-S4 experiment is
$\left\langle\delta T\right\rangle\sim 2\mu K$, leading to $R_{N}\sim 10$
Sugiyama et al. (2017).

Note that considering only the Gaussian contribution of the covariance matrix
(equation (27)) may underestimate the statistical errors. Particularly, the
kSZ effect generally suffers from a correlated non-Gaussian noise due to the
residual foreground contamination, e.g., cosmic infrared background and
thermal SZ effect (see e.g., Bobin et al., 2014; Planck Collaboration et al.,
2018). Though our focus is on relatively large scales and we adopt the
Gaussian covariance, such non-Gaussian contributions need to be taken into
account for a more realistic forecast study.

###  III.4 Fisher matrix formalism

To quantify the constraining power for the dynamical and geometric parameters
above and cosmological parameters, we use the Fisher matrix formalism.
Although forecast studies with the Fisher matrix have been widely performed in
cosmology, there is a limited number of relevant works that consider the kSZ
and IA observations to constrain cosmology, specifically through the RSD and
AP effect. One is the paper by Sugiyama, Okumura & Spergel Sugiyama et al.
(2017), who discussed a benefit of using kSZ observations. Another paper is
Taruya & Okumura Taruya and Okumura (2020), who demonstrated that combining
galaxy clustering with IA observations is beneficial and improves geometric
and dynamical constraints. The present paper complements these two previous
works, and further put forward the forecast study by combining all three
probes.

Given a set of parameters to be estimated, $\\{\theta_{\alpha}\\}$, where
$\alpha=1,\cdots,N_{\theta}$, and provided a set of observed power spectra
$\\{P_{a}\\}$, the Fisher matrix is evaluated with

$\displaystyle F_{\alpha\beta}=$
$\displaystyle\frac{V_{s}}{4\pi^{2}}\int^{k_{\rm max}}_{k_{\rm
min}}dkk^{2}\int^{1}_{-1}d\mu$
$\displaystyle\times\sum_{a,b=1}^{N_{P}}\frac{\partial
P_{a}(k,\mu)}{\partial\theta_{\alpha}}\left[{\rm
Cov}^{-1}\right]_{ab}\frac{\partial P_{b}(k,\mu)}{\partial\theta_{\beta}},$
(32)

where $V_{s}$ is the comoving survey volume for a given redshift range,
$z_{\min}\leq z\leq z_{\max}$, and $k_{\rm min}$ and $k_{\rm max}$ are
respectively the minimum and maximum wavenumbers used for cosmological data
analysis, the former of which is specified with the survey volume by $k_{\rm
min}=2\pi/V_{s}^{1/3}$. Note that for the analysis using a single probe
($N_{P}=1$), namely when we consider either of $P_{gg}$, $P_{EE}$ or $P_{vv}$,
the covariance matrix ${\rm Cov}_{ab}$ is reduced to the power spectrum
squared (see equation (27)).

Provided the Fisher matrix, the expected errors on the parameters of interest,
marginalizing over other parameters, are computed by inverting the Fisher
matrix and constructing the
$\overline{N}_{\vartheta}\times\overline{N}_{\vartheta}$ submatrix
$\overline{\bm{F}}$; for example, when one wants to evaluate the two-
dimensional error contours for a specific pair of parameters,
$\vartheta_{A}\in\,\theta_{\alpha}$ $(A=1,2)$, the $2\times 2$ submatrix is
constructed with $C_{AB}\equiv[\overline{\bm{F}}]^{-1}_{AB}$ $(A,B=1,2)$.
Also, the one-dimensional marginalized error on a parameter $\vartheta_{A}$ is
obtained from $\sigma_{A}^{2}\equiv[\overline{\bm{F}}]^{-1}_{AA}$ (see, e.g.,
Ref. Seo and Eisenstein (2003) for details).

Although our original Fisher matrix is given for the parameters
$\\{\theta_{\alpha}\\}$ determined from the AP effect and RSD, the model-
independent geometric and dynamical constraints are translated into specific
cosmological model constraints by projecting the matrix into a new parameter
space of interest,

$S_{nm}=\sum_{\alpha,\beta}^{N_{\theta}}\frac{\partial\theta_{\alpha}}{\partial
q_{n}}F_{\alpha\beta}\frac{\partial\theta_{\beta}}{\partial q_{m}},$ (33)

where $\\{q_{n}\\}$ is the set of parameters in the new parameter space
($n=1,\cdots,N_{q}$), i.e., non-flat $w_{0}w_{a}\gamma$ CDM model and others
in our case (see section IV.2), and $\bm{S}$ is thus a $N_{q}\times N_{q}$
matrix. Once again, the uncertainties of the parameters can be obtained by
taking the submatrix, e.g., $C_{AB}\equiv[\overline{\bm{S}}]^{-1}_{AB}$,
$\sigma_{A}^{2}\equiv[\overline{\bm{S}}]^{-1}_{AA}$, etc.

For a further discussion on the performance of the constraining power on
multiple parameters, we compute the Figure-of-merit (FoM) defined by

${\rm FoM}=\left\\{{\rm
det}(\overline{\bm{F}})\right\\}^{1/\overline{N}_{\vartheta}},\ \ \ \ \ \ \
{\rm FoM}=\left\\{{\rm
det}(\overline{\bm{S}})\right\\}^{1/\overline{N}_{\varrho}},$ (34)

where quantities with the bar, $\overline{\bm{F}}$ and $\overline{\bm{S}}$,
denote $\overline{N}_{\vartheta}\times\overline{N}_{\vartheta}$ and
$\overline{N}_{\varrho}\times\overline{N}_{\varrho}$ submatrices of $\bm{F}$
and $\bm{S}$
($\overline{N}_{\vartheta}<N_{\theta},\overline{N}_{\varrho}<N_{q}$),
respectively, constructed through the inversion described above. In the
definition provided in Ref. Albrecht et al. (2006),
$\overline{N}_{\vartheta}=\overline{N}_{\varrho}=2$ and the obtained FoM
describes the inverse of the area of the error contour in the marginalized
parameter plane for two parameters. Here, the FoM is defined for an arbitrary
number of parameters, and the obtained value corresponds to a mean radius of
the $\overline{N}_{\vartheta}$ (or $\overline{N}_{\varrho}$) dimensional
volume of the errors.

Table 2: Expected volume, number density and bias of emission line galaxies for given redshift ranges, $z_{\min}\leq z\leq z_{\max}$ of the deep (PFS-like) survey, taken from Ref. Takada et al. (2014). Redshift | Volume $V_{s}$ | $10^{4}n$ | Bias
---|---|---|---
$z_{\rm min}$ | $z_{\rm max}$ | ($h^{-3}{\rm Gpc}^{3}$) | $(h^{3}{\rm Mpc}^{-3})$ | $b_{g}$
0.6 | 0.8 | $0.59$ | $1.9$ | $1.18$
0.8 | 1.0 | $0.79$ | $6.0$ | $1.26$
1.0 | 1.2 | $0.96$ | $5.8$ | $1.34$
1.2 | 1.4 | $1.09$ | $7.8$ | $1.42$
1.4 | 1.6 | $1.19$ | $5.5$ | $1.50$
1.6 | 2.0 | $2.58$ | $3.1$ | $1.62$
2.0 | 2.4 | $2.71$ | $2.7$ | $1.78$

##  IV Results

In this section, we present geometric and dynamical constraints on
cosmological parameters based on the Fisher matrix analysis of galaxy
clustering, IA and kSZ effects. In figure 1, we summarizes the steps of the
analysis of this section graphically, motivated by figure 2 of Ref. Seo and
Eisenstein (2003).

###  IV.1 Setup

To jointly analyze the galaxy clustering, IA and kSZ, we need to use data from
galaxy surveys and CMB experiments: positions and shapes of galaxies are
respectively used to quantify clustering and IA from a galaxy survey, while
the velocity field is inferred by observing the CMB temperature distortion at
the angular position of each galaxy.

Table 3:  Same as table 2 but for the wide (Euclid-like) survey, taken from Ref. Euclid Collaboration et al. (2020). Redshift | Volume $V_{s}$ | $10^{4}n$ | Bias
---|---|---|---
$z_{\rm min}$ | $z_{\rm max}$ | ($h^{-3}{\rm Gpc}^{3}$) | $(h^{3}{\rm Mpc}^{-3})$ | $b_{g}$
0.9 | 1.1 | $7.94$ | $6.86$ | $1.46$
1.1 | 1.3 | $9.15$ | $5.58$ | $1.61$
1.3 | 1.5 | $10.05$ | $4.21$ | $1.75$
1.5 | 1.8 | $16.22$ | $2.61$ | $1.90$

Figure 2:  Two-dimensional $1\sigma$ error contours on the geometric
distances, $D_{\rm A}(z)$ and $H(z)$, and the linear growth rate
$f(z)\sigma_{8}(z)$, expected from the wide (PFS-like) survey. Since there are
seven redshift bins, we here show the result for the central redshift bin,
$1.2<z<1.4$, as an example. Left panel: Constraints from each of the
clustering, kSZ and IA, and the combination of the latter two. Right panel:
Similar to the left panel but joint constraints from the combination of kSZ,
IA and galaxy clustering. Note that the joint constraints obtained from
clustering and IA (red contours) almost overlap with those from clustering, IA
and kSZ (blue contours).

As we mentioned in section I, there are a number of planned spectroscopic
galaxy surveys aiming at constraining cosmology with a high precision. These
surveys are generally categorized into the two types: (narrow but) deep
surveys and (shallow but) wide surveys. In the Fisher matrix analysis below,
we consider the Subaru PFS and Euclid as examples of deep and wide surveys,
respectively, both of which target emission line galaxies (ELG) as a tracer of
the LSS. Tables 2 and 3 show the redshift range, survey volume, and number
density and bias of the ELG samples for the PFS Takada et al. (2014) and
Euclid Euclid Collaboration et al. (2020), respectively. Ref. Shi et al.
(2021) has proposed an estimator to directly detect IA of host halos using the
observation of the ELGs. In the forecast analysis presented below, we consider
that the power spectra related to the IA are measured with this
estimator.222Even though we use elliptical galaxies as a tracer of the tidal
field as in the conventional analysis, we can present a similar analysis based
on luminous red galaxy samples from, i.e., DESI, and the main results below
will not change qualitatively (e.g., Taruya and Okumura (2020)). Following the
result of Ref. Shi et al. (2021), we set the fiducial value of the IA
amplitude to $A_{\rm IA}=18$, assuming its redshift independence. The PFS
galaxy sample provides high-quality shape information thanks to the imaging
survey of the HSC Miyazaki et al. (2012); Aihara et al. (2018), and we thus
set the shape noise, $\sigma_{\gamma}$, to $\sigma_{\gamma}=0.2$ for the deep
survey Hikage et al. (2019). For the wide survey, following Ref. Euclid
Collaboration et al. (2020), we set it to $\sigma_{\gamma}=0.3$. We will
discuss the effect of changing the fiducial values of $A_{\rm IA}$ and
$\sigma_{\gamma}$ in section V.

Similarly to the forecast study of the kSZ effect in Ref. Sugiyama et al.
(2017), we consider CMB-S4 Abazajian et al. (2016) as a CMB experiment for the
expected observation of the kSZ effect. While the angular area of the PFS is
completely overlapped with that of the CMB-S4, the half of the Euclid area is
covered by the CMB-S4 Euclid Collaboration et al. (2022). Thus, when
considering the statistics related to the kSZ effect, namely $P_{vv}$,
$P_{gv}$ and $P_{vE}$, in the wide survey, the elements of the covariance
matrix for these statistics are multiplied by two. Furthermore, the values of
$k_{\min}$ for these terms become larger by the factor of $2^{1/3}$. We choose
$R_{\rm N}=10$ as our fiducial choice, following Ref. Sugiyama et al. (2017).
For the velocity dispersion, we use the liner theory value as a fiducial
value, $\sigma_{v}=\sigma_{v,{\rm lin}}$. The combination of $(1+R_{\rm
N}^{2})\sigma_{v}^{2}$ contributes to the shot noise of the kSZ power
spectrum. We will test the effect of these choices in section V.

In the following analysis, we assume the spatially flat $\Lambda$CDM model as
our fiducial model Planck Collaboration et al. (2016a): $\Omega_{\rm
m}=1-\Omega_{\rm DE}=0.315$, $\Omega_{K}=0$, $w_{0}=-1$, $w_{a}=0$,
$H_{0}=67.3~{}[{\rm km/s/Mpc}]$ and the present-day value of $\sigma_{8}$,
$\sigma_{8,0}\equiv\sigma_{8}(z=0)$, to be $\sigma_{8,0}=0.8309$. For
computation of the linear power spectrum in equation (19), $P_{\rm lin}(k;z)$,
we use the publicly-available CAMB code Lewis et al. (2000). When we consider
the model which allows deviation of the structure growth from GR prediction,
we set the fiducial value of $\gamma$ in equation (11) to be consistent with
GR, $\gamma=0.545$.

Finally, the maximum wavenumber of the power spectra used for the cosmological
analysis with the Fisher matrix is set to $k_{\rm max}=0.2\,h$ Mpc-1. While
forecast results with this choice, presented below as our main results, give
tight geometrical and dynamical constraints, we also consider in Appendix B a
conservative choice of $k_{\rm max}=0.1\,h\,{\rm Mpc}^{-1}$, and discuss its
impact on the parameter constraints.

Figure 3:  Left set: one-dimensional marginalized errors on the growth rate
$f(z)\sigma_{8}(z)$ (upper panel) and geometric distances (lower panel),
$D_{A}(z)$ and $H(z)$, expected from the deep (PFS-like) survey. Right set:
same as the left set but the result expected from wide (Euclid-like) survey.
The errors on $H$ and $D_{\rm A}$ are multiplied by 4 and 8, respectively, for
illustration.

###  IV.2 Geometric and dynamical constraints

Let us first look at model-independent dynamical and geometric constraints,
namely the constraints on $f(z)\sigma_{8}(z)$, $D_{\rm A}(z)$ and $H(z)$,
expected from the upcoming Subaru PFS survey. From the original Fisher matrix
which includes these parameters in addition to nuisance parameters, as
summarized in table 1, we obtain the marginalized constraints as described in
section III.4.

The left panel of figure 2 shows the two-dimensional $1\sigma$ error contours
on $f\sigma_{8}$, $D_{\rm A}$ and $H$ normalized by their fiducial values,
which are obtained individually from galaxy clustering ($P_{gg}$), kSZ
($P_{vv}$) and IA ($P_{EE}$). Since the PFS is a deep survey and has seven
redshift bins at $0.6<z<2.4$ (see Table 2), we here plot the result for the
central redshift bin, $1.2<z<1.4$, where the number density of galaxies is the
largest. Note that the left panel of figure 2 does not consider any cross
correlation between different probes, namely $P_{gE}$, $P_{gv}$ and $P_{vE}$
(see table 1). As clearly shown in the figure, using either $P_{EE}$ or
$P_{vv}$ cannot constrain the growth rate. This is because the intrinsic
galaxy shapes themselves are insensitive to RSD in linear theory and the kSZ
only constrains the combination of $f\sigma_{8}$ and $\tau$ without imposing
any prior on $\tau$. Nevertheless, each single measurement of kSZ and IA can
give meaningful constraints on $D_{\rm A}$ and $H$. Then, including the cross
correlation, the combination of the two probes, namely $P_{vv}$ and $P_{EE}$
as well as $P_{vE}$, improves the constraint on $(D_{\rm A},H)$, depicted as
the blue contour.

Interestingly, the constraining power on $D_{\rm A}$ and $H$, when combining
kSZ and IA, can become tighter, and for the one-dimensional marginalized
error, the precision on each parameter achieves a few percent level. Although
the galaxy clustering still outperforms the kSZ and IA observations,
systematic effects in each probe come to play differently (e.g., galaxy bias,
shape noises and optical depth), and in this respect, the geometric
constraints from the kSZ and IA are complementary as alternatives to those
from the galaxy clustering. Thus, constraining the geometric distances with
kSZ and/or IA effects would help addressing recent systematics-related issues
such as the Hubble tension.

The right panel of figure 2 shows the result similar to the left panel, but
the joint constraints combining kSZ and/or IA with galaxy clustering. Compared
to the results from the single probe, the constraints are indeed improved, as
previously demonstrated in Refs. Sugiyama et al. (2017) (clustering+kSZ) and
Taruya and Okumura (2020) (clustering+IA). Here we newly show that the
combination of all three probes, characterized by the six power spectra, can
further tighten the constraints on both the geometric distances and growth of
structure. The results imply that adding any of these power spectra can
extract independent cosmological information even though they measure the same
underlying matter field. The left panel of figure 3 summarizes the one-
dimensional marginalized errors on $f\sigma_{8}$, $D_{\rm A}$ and $H$ expected
from the deep (PFS-like) survey, plotted as a function of $z$ over
$0.6<z<2.4$. Over all redshifts studied here, adding the information from kSZ
and IA measurements does improve the geometric and dynamical constraints.

Table 4: Summary of the cosmological models investigated in the forecast study. Model | No. of free | Parameters $\\{q_{n}\\}$ | CMB prior | Result
---|---|---|---|---
| parameters $N_{q}$ | Flat | Non-flat | MG |  | (Fig.)
$w_{0}$ flat | 4 | $\Omega_{\rm m},H_{0},w_{0},\sigma_{8}$ | $-$ | $-$ | $-$ | 4
$w_{0}w_{a}$ flat | 5 | $\Omega_{\rm m},H_{0},w_{0},w_{a},\sigma_{8}$ | $-$ | $-$ | Yes | 5
$w_{0}w_{a}$ non-flat | 6 | $\Omega_{\rm m},H_{0},w_{0},w_{a},\sigma_{8}$ | $\Omega_{K}$ | $-$ | Yes | 6
$w_{0}w_{a}\gamma$ flat | 6 | $\Omega_{\rm m},H_{0},w_{0},w_{a},\sigma_{8}$ | $-$ | $\gamma$ | Yes | 7
$w_{0}w_{a}\gamma$ non-flat | 7 | $\Omega_{\rm m},H_{0},w_{0},w_{a},\sigma_{8}$ | $\Omega_{K}$ | $\gamma$ | Yes | 8

###  IV.3 Cosmological parameter constraints

Figure 4: Cosmological constraints on the $w_{0}$ flat model expected from the
deep (PFS-like) survey without relying on the CMB prior. In each panel,
contours show the $1\sigma$ confidence levels, with the amplitude parameter
today, $\sigma_{8,0}=\sigma_{8}(0)$, marginalized over. Note that the joint
constraints obtained from clustering and IA (red contours) are almost entirely
behind those from clustering, IA and kSZ (blue contours).  Figure 5:
Cosmological constraints on the $w_{0}w_{a}$ flat model expected from the deep
(PFS-like) survey. Unlike figure 4, CMB prior information is added here. In
each panel, contours show the $1\sigma$ confidence levels, with the amplitude
parameter today, $\sigma_{8,0}=\sigma_{8}(0)$, marginalized over. The joint
constraints obtained from clustering and IA (red contours) are almost entirely
behind those from clustering, IA and kSZ (blue contours).

Provided the model-independent geometric and dynamical constraints estimated
from the original Fisher matrix in section IV.2, we further discuss specific
cosmological model constraints listed in Table 4. In what follows, except the
$w_{0}$ flat CDM model, we add the CMB prior information to constrain
cosmological parameters and follow the conventional approach adopted in the
data analysis of BOSS Eisenstein et al. (2005); Percival et al. (2007);
Beutler et al. (2017), which do not use the information of the full-shape
power spectra. To be precise, we introduce the following scaling parameters:

$\alpha_{\parallel}=\frac{H(z)r_{d}}{H^{\rm fid}(z)r_{d}^{\rm fid}},\ \ \ \ \
\ \alpha_{\perp}=\frac{D_{\rm A}(z)r_{d}^{\rm fid}}{D_{\rm A}^{{\rm
fid}}(z)r_{d}},$ (35)

where the quantity $r_{d}$ is the sound horizon scale at the drag epoch
$z_{d}$ when photons and baryons are decoupled Eisenstein and Hu (1998), given
by

$r_{d}=\int^{\infty}_{z_{d}}\frac{c_{s}(z)}{H(z)}dz,$ (36)

with $c_{s}$ being the sound speed in the photon-baryon fluid. We then
redefine the fiducial wavenumbers $k_{\parallel}$ and $k_{\perp}$, which
appear in Eq. (20), as $k_{\parallel}^{\rm
fid}=k_{\parallel}/\alpha_{\parallel}$ and $k_{\perp}^{\rm
fid}=k_{\perp}\alpha_{\perp}$. With this parameterization, the original
expression for the power spectrum at Eq. (20), taking the AP effect into
account, is recast as

$P_{ij}^{{\rm obs}}\left(k_{\perp}^{\rm fid},k_{\parallel}^{{\rm
fid}};z\right)=\left(\frac{r_{d}^{\rm
fid}}{r_{d}}\right)^{3}\frac{\alpha_{\parallel}}{\alpha_{\perp}^{2}}P_{ij}\left(k_{\perp},k_{\parallel};z\right),$
(37)

where the prefactor $(r_{d}^{\rm
fid}/r_{d})^{3}(\alpha_{\parallel}/\alpha_{\perp}^{2})$ is equivalent to that
in equation (20). Note that the dimensionless quantities $r_{d}/D_{\rm A}$ and
$Hr_{d}$ are related to the actual BAO scales measurable from galaxy surveys,
i.e., angular separation and redshift width of the acoustic scales. In this
respect, with the form given in equation (37), we are assuming that the main
contribution to the AP effect comes from the BAO. As discussed in Ref. Beutler
et al. (2017), the uncertainty on the $r_{d}$ measurement from the Planck
experiment is only at the level of $\sim 0.2$ per cent Planck Collaboration et
al. (2016a) and fixing $r_{d}$ in equation (37) has a negligible effect on our
cosmological parameter estimation. Based on this argument, we approximately
set the pre-factor $(r_{d}^{\rm fid}/r_{d})^{3}$ to unity for the Fisher
matrix analysis below.

Figure 6:  Same as figure 5 but for the $w_{0}w_{a}$ non-flat model.  Figure
7:  Same as figure 5 but for the $w_{0}w_{a}\gamma$ flat model.

Figure 8:  Top: Same as figure 5 but for the $w_{0}w_{a}\gamma$ non-flat model from the deep (PFS-like) survey. Bottom: Similar to the top panel but from the wide (Euclid-like) survey.  Table 5: Fractional marginalized errors on cosmological parameters, $\sigma/\theta^{\rm fid}$, for the four specific models. The CMB prior information is added for all the results here. Since the fiducial values of $w_{a}$ and $\Omega_{K}$ are zero, we show the absolute errors, $\sigma$. Since the absolute errors on $\Omega_{K}$ are small, we show the errors multiplied by 100. |  |  | Deep (PFS-like) survey |  | Wide (Euclid-like) survey
---|---|---|---|---|---
|  |  | Clustering |  |  |  |  | Clustering |  |  |
Model | $\sigma/\theta^{\rm fid}$ |  | only | +kSZ | +IA | +IA+kSZ |  | only | +kSZ | +IA | +IA+kSZ
$w_{0},w_{a}$ | $\Omega_{\rm m}$ |  | 0.0850 | 0.0813 | 0.0765 | 0.0746 |  | 0.0590 | 0.0578 | 0.0559 | 0.0555
flat | $w_{0}$ |  | 0.230 | 0.224 | 0.213 | 0.208 |  | 0.164 | 0.163 | 0.157 | 0.156
| $w_{a}$ |  | 0.638 | 0.613 | 0.584 | 0.569 |  | 0.455 | 0.450 | 0.438 | 0.434
| $H_{0}$ |  | 0.0383 | 0.0363 | 0.0338 | 0.0329 |  | 0.0257 | 0.0251 | 0.0241 | 0.0239
$w_{0},w_{a}$ | $\Omega_{\rm m}$ |  | 0.0877 | 0.0841 | 0.0788 | 0.0767 |  | 0.0592 | 0.0580 | 0.0561 | 0.0556
non-flat | $w_{0}$ |  | 0.240 | 0.232 | 0.220 | 0.214 |  | 0.164 | 0.163 | 0.157 | 0.156
| $w_{a}$ |  | 0.665 | 0.634 | 0.600 | 0.583 |  | 0.457 | 0.452 | 0.440 | 0.437
| $H_{0}$ |  | 0.0415 | 0.0397 | 0.0370 | 0.0360 |  | 0.0262 | 0.0256 | 0.0247 | 0.0244
| $100\Omega_{K}$ |  | 0.231 | 0.229 | 0.223 | 0.222 |  | 0.162 | 0.161 | 0.161 | 0.161
$w_{0},w_{a},\gamma$ | $\Omega_{\rm m}$ |  | 0.1459 | 0.1244 | 0.1075 | 0.0990 |  | 0.1004 | 0.0955 | 0.0840 | 0.0804
flat | $w_{0}$ |  | 0.415 | 0.359 | 0.314 | 0.288 |  | 0.288 | 0.272 | 0.245 | 0.234
| $w_{a}$ |  | 1.036 | 0.907 | 0.804 | 0.743 |  | 0.761 | 0.715 | 0.655 | 0.625
| $H_{0}$ |  | 0.0691 | 0.0585 | 0.0501 | 0.0458 |  | 0.0452 | 0.0429 | 0.0374 | 0.0358
| $\gamma$ |  | 0.271 | 0.238 | 0.217 | 0.202 |  | 0.193 | 0.182 | 0.169 | 0.161
$w_{0},w_{a},\gamma$ | $\Omega_{\rm m}$ |  | 0.1679 | 0.1380 | 0.1162 | 0.1055 |  | 0.1114 | 0.1039 | 0.0884 | 0.0836
non-flat | $w_{0}$ |  | 0.484 | 0.400 | 0.340 | 0.307 |  | 0.330 | 0.304 | 0.264 | 0.248
| $w_{a}$ |  | 1.202 | 1.004 | 0.865 | 0.786 |  | 0.841 | 0.774 | 0.686 | 0.646
| $H_{0}$ |  | 0.0833 | 0.0682 | 0.0571 | 0.0516 |  | 0.0548 | 0.0510 | 0.0430 | 0.0404
| $100\Omega_{K}$ |  | 0.258 | 0.245 | 0.234 | 0.230 |  | 0.194 | 0.189 | 0.182 | 0.179
| $\gamma$ |  | 0.304 | 0.256 | 0.228 | 0.210 |  | 0.231 | 0.212 | 0.191 | 0.179

Now, the model-independent parameters in our original Fisher matrix, combining
all three probes, become $\theta_{\alpha}=(b\sigma_{8},A_{\rm
IA}\sigma_{8},\tau,f\sigma_{8},\alpha_{\parallel},\alpha_{\perp})$, and the
marginalized constraints on
$\vartheta_{A}=(f\sigma_{8},\alpha_{\perp},\alpha_{\parallel})$ are evaluated
for each $z$-slice by constructing the $3\times 3$ sub-matrix
$\overline{\bm{F}}_{\rm LSS}(z_{k})$. Summing up these sub-matrices over all
the redshift bins, i.e., $\overline{\bm{F}}_{{\rm
LSS}}=\sum_{k}\overline{\bm{F}}_{{\rm LSS}}(z_{k})$, we project it into a new
parameter space to test the model-dependent cosmological parameters $q_{n}$
through equation (33). The most general model considered in our analysis is
the $w_{0}w_{a}\gamma$ non-flat model, with $q_{n}=(\Omega_{\rm
m},w_{0},w_{a},H_{0},\Omega_{K},\gamma,\sigma_{8,0})$. All the cosmological
models we consider in this paper are summarized in table 4.

Let us show our main results for the deep, PFS-like survey below. Figures 4 –
7 and the top panel of figure 8 plot the expected two-dimensional constraints
on pairs of model parameters for different cosmological models. Also, table 5
and figure 9 summarize the one-dimensional marginalized constraints. We will
discuss all the results in detail in the rest of this subsection. Except for
figure 4, all the following results are obtained adding the CMB prior
information, as detailed in Appendix A. Thus, the constraints are obtained
from the combination of the Fisher matrices of the LSS and CMB,
$\bm{S}=\bm{S}_{\rm LSS}+\bm{S}_{\rm CMB}$. For all cases, the nuisance
parameters characterizing the power spectrum normalization on each probe
namely $b\sigma_{8}$, $\tau$, and $A_{\rm IA}\sigma_{8}$, are marginalized
over. Comparisons of the obtained constraints with those from the wide,
Euclid-like survey will be presented in section V.1.

Figure 9: Marginalized $1\sigma$ errors on cosmological parameters, relative
to their corresponding fiducial values, $\sigma/\theta^{\rm fid}$. The darkly
and lightly filled bars show the errors from the deep (PFS-like) and wide
(Euclid-like) surveys, respectively. The top-left and top-right panels are for
the $w_{0}w_{a}$ flat and non-flat models, respectively. Similarly, the
bottom-left and bottom-right panels are for $w_{0}w_{a}\gamma$ flat and non-
flat models, respectively. The CMB prior information is added for all the
results here. Since the fiducial values of $w_{a}$ and $\Omega_{K}$ are zero,
we show the absolute errors, $\sigma$. Since the absolute errors on
$\Omega_{K}$ are small, we show the errors multiplied by 100. The hollow bars
are similar with the filled bars but based on the conservative analysis with
the scales of $k\leq 0.1\,h\,{\rm Mpc}^{-1}$ (see appendix B).

Figure 4 shows the case for the $w_{0}$ flat model, in which we vary
$q_{n}=(\Omega_{m},w_{0},H_{0},\sigma_{8,0})$. Only for this model, we do not
add the CMB prior and use LSS probes as our primary data set. As shown in Ref.
Taruya and Okumura (2020), adding IA to galaxy clustering significantly
improves the constraints. If the kSZ measurement is added, one can achieve a
similar (but slightly weaker) improvement. Simultaneously analyzing galaxy
clustering with kSZ and IA, the constraint on each cosmological parameter gets
even tighter, by $15-21\%$, compared to the clustering-only constraints.

In figure 5, adding the CMB prior information, we show an extension of the
parameter space by allowing the time-varying dark energy equation-of-state,
which is the $w_{0}w_{a}$ flat model described by the parameters
$q_{n}=(\Omega_{\rm m},w_{0},w_{a},H_{0},\sigma_{8,0})$. Here, the improvement
by adding IA is not so significant compared to the former case, due mainly to
a dominant contribution from the CMB prior, consistent with the result of Ref.
Taruya and Okumura (2020). However, combining the galaxy clustering with both
kSZ and IA measurements, we can improve the constraints further, for example,
on $w_{a}$ by $\sim 11\%$, as shown in table 5 and figure 9. Figure 6 examines
the case with non-zero $\Omega_{K}$, by introducing another degree of freedom
in the parameter space on top of the $w_{0}w_{a}$ flat model. Note that based
on the BAO experiments at high $z$, a best achievable precision on
$\Omega_{K}$, limited by the cosmic variance, has been studied in detail in
Ref. Takada and Doré (2015). In our forecast, the spatial curvature has
already been tightly constrained by the CMB prior. Thus, the resulting
constraints are similar with those of the model with $\Omega_{K}=0$ in figure
5.

Figure 10:  FoM of cosmological impact for clustering only (gray), clustering
$+$ kSZ (yellow), clustering $+$ IA (red) and clustering $+$ IA $+$ kSZ
(blue). The upper-left and upper-right panels are the results for the
$w_{0}w_{a}$ flat and non-flat models, respectively. Similarly, the lower-left
and lower-right panels are the results for the $w_{0}w_{a}\gamma$ flat and
non-flat models, respectively. The CMB prior information is added for all the
results here. In each panel, the values of FoM are normalized by that for the
PFS survey with clustering-only analysis. The yellow, red and blue vertical
lines indicate the FoM values obtained in the upper-left panel for comparison.

Figure 11: Relative impact of combining IA (left set) and kSZ (right set) on
the parameter constraints, defined by the ratio of figure-of-merit, ${\rm
FoM}_{g+E}/{\rm FoM}_{g}$ and ${\rm FoM}_{g+v}/{\rm FoM}_{g}$, respectively.
The subscripts of $g$, $g+E$ and $g+v$ denote the FoM expected from galaxy
clustering only, the combination of clustering and IA, and that of clustering
and kSZ, respectively. Upper panels show the results for geometric distances
and structure growth, $D_{\rm A}$, $H$, $f\sigma_{8}$, derived from each
redshift slice of the PFS survey. Results at $z=1.1$ and $z=1.5$ are not shown
here because they are almost equivalent to those at $z=0.9$. Bottom panels are
the results for cosmological parameters, with $\sigma_{8,0}$ marginalized
over. In all cases, the vertical dashed lines indicate the default parameter
setup (see Sec. IV.1).

Now, allowing the deviation of growth of structure from the GR prediction,
characterized by the parameter $\gamma$, we test and constrain both the cosmic
expansion and gravity law, shown in figure 7 and the top panel of figure 8.
Figure 7 considers the $w_{0}w_{a}\gamma$ flat model, in which the spatial
curvature is kept flat. The resulting constraints from the clustering-only
analysis are generally weaker than the case of $w_{0}w_{a}$ non-flat model
despite the fact that the number of parameters remains unchanged. The main
reason comes from the newly introduced parameter $\gamma$, which can be
constrained only through the measurement of the growth rate, and is strongly
degenerated with $\Omega_{\rm m}$. Nevertheless, adding the information from
the observations of kSZ and/or IA, the constraints get significantly tighter,
and combining all three probes, the achievable precision is improved by $25\%$
for $\gamma$, and $\sim 30\%$ for other parameters, as shown in table 5 and
figure 9. In the top panel of figure 8, the significance of combining all
three probes is further enhanced in $w_{0}w_{a}\gamma$ non-flat model, where
we have seven parameters of $q_{n}=(\Omega_{\rm
m},\Omega_{K},w_{0},w_{a},H_{0},\gamma,\sigma_{8,0})$. As a result, compared
to the clustering-only analysis, the simultaneous analysis with the
clustering, IA and kSZ further improves the constraints by $31\%$ for $\gamma$
and $>35\%$ for others except for $\Omega_{K}$ (see table 5 and figure 9).

##  V Discussion

###  V.1 Deep vs wide surveys

So far, we have considered the PFS survey as a representative example of deep
galaxy surveys. Here, we discuss how the constraining power of kSZ and IA
measurements depends on types of galaxy surveys. For this purpose, we perform
the forecast analysis for the Euclid survey as an example of wide galaxy
survey. The right panel of figure 3 presents geometric and dynamical
constraints from the Euclid-like survey. Though the redshift range for the
Euclid is narrower than that for the PFS, the constraints on $f\sigma_{8}$,
$D_{\rm A}$ and $H$ at each redshift bin are much tighter due to the large
survey volumes (see table 3). Cosmological constraints are thus expected to be
stronger as well. To see it quantitatively, let us utilize the FoM introduced
in equation (34). Here, we marginalize over the amplitude parameter today,
$\sigma_{8,0}$, via the inversion of the $N_{q}\times N_{q}$ Fisher matrix,
$\bm{S}$ (see equation (33)). The size of the matrix $\overline{\bm{S}}$ is
thus ($N_{q}-1)\times(N_{q}-1)$. Indeed, the FoM for cosmological parameters
from the wide survey is always better, roughly by a factor of two, than that
from the PFS. The comparison is shown for the four cosmological models in
figure 10.

Constraints on each cosmological parameter is made with the projection of the
Fisher matrix. The forecast results from the Euclid survey are summarized in
the right hand side of table 5 and figure 9. If one uses only the information
of clustering, constraints from the wide survey considered here are always
tighter than those from the deep survey, by $25-40\%$. Then one can improve
the constraints by the joint analysis of clustering, IA and kSZ, similarly to
the analysis of deep galaxy surveys. However, the improvement of the
cosmological constraints are not so significant as the case of the deep
survey. It is particularly prominent if we consider the model which allows the
$\gamma$ parameter to vary. For example, in the $w_{0}w_{a}\gamma$ flat model,
while the improvement of cosmological parameters for the deep survey is
$25-34\%$, that for the wide survey is $17-21\%$. It could be due to the fact
that the $\gamma$ parameter is constrained from the redshift dependence of the
measured growth rate $f(z)$ at various redshifts, and thus the constraining
power in the wide survey does not gain as much as that in a deep survey by
combining with additional probes of kSZ and IA. As a result, if we perform a
joint analysis of galaxy clustering together with kSZ and IA for a deep
survey, the constraining power can be as strong as the conventional
clustering-only analysis for a wide survey even though the FoM for the wide
survey is twice as large. More interestingly, in the most general
$w_{0}w_{a}\gamma$ non-flat model, even the deep survey with the combination
of IA and clustering can have the constraining power as strong as the the wide
survey, as shown in table 5 and figure 9. If one combines all the three probes
in the deep survey, the constraints become stronger than the conventional
clustering analysis in the wide survey. We also show the two-dimensional error
contours of the cosmological parameters from the wide survey in the bottom
panel of figure 8 which can be compared to those from the deep survey in the
top panel. These results clearly demonstrate the importance of considering the
IA and kSZ effects.

###  V.2 Choices of fiducial survey parameters

The results of our Fisher matrix analyses in section IV and V.1 rely on the
specific setup based on the upcoming surveys. Among several potential concerns
in the actual observations, the expected amplitude and error of kSZ and IA
statistics are less certain than those of galaxy clustering. Specifically, the
benefit of the IA statistics largely depends on the fiducial setup of the
parameters $\sigma_{\gamma}$ and $A_{\rm IA}$, while that of the kSZ
statistics is affected by the choice of $\sigma_{v}$ and $R_{N}$. In this
subsection, we discuss the robustness of the benefit combining the IA and kSZ
data set with the galaxy clustering. To elucidate this, allowing the
parameters $\sigma_{\gamma}$, $A_{\rm IA}$, and $R_{N}$ (or $\sigma_{v}$) to
vary, we estimate the FoM, defined by equation (34).

Figure 11 shows the ratio of the FoM for the combined data set of galaxy
clustering and IA (or kSZ) to that for the galaxy clustering alone, ${\rm
FoM}_{g+E}/{\rm FoM}_{g}$ (or ${\rm FoM}_{g+v}/{\rm FoM}_{g}$). The rightmost
panels of the figure will be discussed in the next subsection. The upper
panels plot the results for the geometric and dynamical constraints, i.e.,
$D_{\rm A}$, $H$ and $f\sigma_{8}$ at each redshift slice. On the other hand,
lower panels show the FoM for the cosmological parameters. As seen in the
upper panels, the benefit of combining kSZ and/or IA statistics increases with
the number density of galaxies, e.g., $10^{4}n=1.9,6.0,7.8,3.1$, and $2.7$
$[h^{3}{\rm Mpc}^{-3}]$ at $z=0.7,0.9,1.3,1.8$, and $2.2$ (see Table 2). Note
that the results in the lower panels are obtained by adding the CMB prior
information, with the fluctuation amplitude, $\sigma_{8,0}$, marginalized
over. Thus, the number of cosmological parameters used to compute the FoM in
equation (34) is $N_{p}=4,5,5$ and $6$ for the red, green, blue and yellow
curves, respectively. As expected from the results in section IV.3, the impact
of combining IA or kSZ on the improvement of cosmological parameters is more
significant for the models varying $\gamma$ and less significant for that
varying $\Omega_{K}$. Even with the suppressed amplitude of
ellipticity/velocity fields or enhanced shape noise by a factor of 2, one can
still expect a fruitful benefit from the combination of galaxy clustering with
IA/kSZ. In particular, adopting the $w_{0}w_{a}\gamma$ non-flat model, the
improvement on each parameter reaches $\sim 20\%$, compared to the case with
galaxy clustering data alone.

###  V.3 Effect of line-of-sight structures on kSZ statistics

In this paper, as in previous works (e.g., Hand et al., 2012), we considered
that the kSZ effect is observed in a three-dimensional space, and statistical
properties of the measured velocity fields are described by the three-
dimensional matter power spectrum through equations (19) with (13). However,
the contribution of the kSZ effect to CMB anisotropies is in general given by
a line-of-sight integral of the velocity field. Thus, unless we use massive
galaxy groups or clusters as a tracer of the velocity field, the measured kSZ
signals would be affected by other velocity components arising predominantly
from diffuse and extended sources that may not fairly trace the large-scale
matter flow, hence leading to a suppression of the three-dimensional power
spectra Hernández-Monteagudo et al. (2015). To see this effect, we approximate
the impact of the line-of-sight integral by introducing a multiplicative
Gaussian smoothing kernel with the typical correlation length $D_{\rm los}$,
$G_{\parallel}(k_{\parallel};D_{\rm los})=e^{-k_{\parallel}^{2}D_{\rm
los}^{2}/2}$. The kSZ distortion field, $\delta T(\bm{k})$, is then modulated
as $\delta T(\bm{k})\rightarrow\delta
T(\bm{k})G_{\parallel}(k_{\parallel};D_{\rm los})$. Accordingly, the power
spectra that include the velocity field, $P_{gv}(\bm{k}),P_{vE}(\bm{k})$ and
$P_{vv}(\bm{k})$, are modulated as
$P_{gv}(\bm{k})G_{\parallel}(k_{\parallel};D_{\rm los})$,
$P_{vE}(\bm{k})G_{\parallel}(k_{\parallel};D_{\rm los})$, and
$P_{vv}(\bm{k})G_{\parallel}^{2}(k_{\parallel};D_{\rm los})$, respectively. It
is not trivial how the line-of-sight structure affects the velocity
dispersion, $\sigma_{d}^{2}$, which appears in the shot noise contribution
(see equation (29)). Although such a structure may introduce additional noise
contribution to the one modeled by equation (31), we assume for simplicity
that the velocity dispersion remains unchanged, as we have already seen the
impact of the increased velocity dispersion on FoM in left panels of figure
11.

The rightmost panels of figure 11 show the ratio of the FoM for the combined
data set of galaxy clustering and kSZ to that for the galaxy clustering alone,
${\rm FoM}_{g+v}/{\rm FoM}_{g}$, as a function of the smearing length, $D_{\rm
los}$. Note that in estimating the FoM, we consider that the damping function
$G$ is not a properly modeled factor, and for a conservative estimate, we do
not take into account the AP effect of this function. The fractional gain of
the FoM by adding kSZ decreases with increasing $D_{\rm los}$, as expected.
However, even with such a conservative setting, we can still expect $5-10\%$
improvements at typical values of $D_{\rm los}$, $D_{\rm los}\sim
40-60~{}h^{-1}\,{\rm Mpc}$.

As another example, let us also consider the case where the velocity
dispersion including the diffuse/extended components is modeled by the line-
of-sight integral just like the kSZ power spectra themselves. In such a case,
the expression of the velocity dispersion in equation (31) is modulated as

$\displaystyle\sigma_{d}^{2}$
$\displaystyle=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{\mu^{2}P_{\theta\theta}(q;z)}{q^{2}}G_{\parallel}^{2}(q\mu;D_{\rm
los})$
$\displaystyle=\frac{1}{6\pi^{2}}\int^{\infty}_{0}dqP_{\theta\theta}(q)\left\\{-\frac{3e^{-q^{2}D_{\rm
los}^{2}}}{2q^{2}D_{\rm los}^{2}}+\frac{3\sqrt{\pi}{\rm erf}(qD_{\rm
los})}{4q^{3}D_{\rm los}^{3}}\right\\}$
$\displaystyle=\frac{1}{6\pi^{2}}\int^{\infty}_{0}dqP_{\theta\theta}(q)\left(1-\frac{3}{5}q^{2}D_{\rm
los}^{2}+\frac{3}{14}q^{4}D_{\rm los}^{4}-\cdots\right),$ (38)

where ${\rm erf}(x)$ is the error function and the third equality is derived
by the Taylor expansion. Adopting the estimation of the velocity dispersion
given above, we find that the fractional gain of adding the kSZ effect is
almost unchanged, a few per cent, at $D_{\rm los}=50h^{-1}\,{\rm Mpc}$,
compared to the undamped case ($D_{\rm los}=0h^{-1}\,{\rm Mpc}$).

Throughout this paper, we have considered the “homogeneous” kSZ effect, which
arises when the reionization process is complete Sunyaev and Zeldovich (1980);
Ostriker and Vishniac (1986). However, on top of that, there is a residual kSZ
effect due to the “patchy” (or inhomogeneous) reionization, which arises
during the process of reionization, from the proper motion of ionized bubbles
around emitting sources, and it can be an additional source of the noise for
the kSZ signal Aghanim et al. (1996). The contribution of the patchy kSZ
effect becomes significant at small scales, $\ell\sim k\chi>1000$ Aghanim et
al. (1996); Planck Collaboration et al. (2016b), while our analysis focuses
the data only up to quasi-nonlinear scales, $k\leq 0.2h\,{\rm Mpc}^{-1}$. Thus
in our analysis we safely ignore this effect. However, when we perform a more
aggressive analysis including higher-$k$ modes, this patchy reionization
effect needs to be properly taken into account.

###  V.4 Contribution of gravitational lensing to IA statistics

So far, we have considered the observation of IA as one of the cosmological
probes ignoring the lensing effect. In principle, the shape of the galaxies,
projected onto the sky, can be very sensitive to the lensing effect, and has
been extensively used to detect and measure the cosmic shear signals. This
implies that unless properly modeling it, the lensing effect on the E-mode
ellipticity may be regarded as a potential systematics that can degrade the
geometric and dynamical constraints. Nevertheless, one important point in the
present analysis using the IA is that, in contrast to the conventional lensing
analysis, one gets access to the cosmological information from the three-
dimensional power spectrum. In this subsection, we discuss a quantitative
impact of the lensing contribution on the observations of IA, particularly
focusing on the three-dimensional power spectrum of E-mode ellipticity.

In the presence of the lensing effect, the observed E-mode ellipticity defined
in the three-dimensional Fourier space, $\gamma_{\rm E}(\bm{k};z)$, is divided
into two pieces, $\gamma_{\rm E}=\gamma_{\rm E}^{\rm(I)}+\gamma_{\rm
E}^{\rm(GL)}$. Here the former is originated from the IA, and the latter
represents the lens-induced ellipticity. Then the (auto) E-mode power spectrum
measured at a redshift $z$ is expressed as

$\displaystyle
P_{EE}(\bm{k};z)=P_{EE}^{\rm(I)}(\bm{k};z)+P_{EE}^{\rm(GL)}(\bm{k};z).$ (39)

Note that in principle, there exists the cross talk between IA and lensing,
i.e., the gravitational shear–intrinsic ellipticity correlation. However, such
a cross talk becomes non-vanishing only if we take the correlation between
different $z$-slices. Since the geometric and dynamical constraints considered
in this paper are obtained from individual $z$-slices, the relevant quantity
to be considered is only the E-mode lensing spectrum, $P_{EE}^{\rm(GL)}$.

Figure 12: Impact of lensing effects on the auto power spectra of E-mode
ellipticity $P_{EE}$ (upper row) and cross power spectra between density and
E-mode ellipticity $P_{gE}$ (lower row), at $z=0.7$ (left), $1.30$ (middle),
and $2.20$ (right), which are the lowest, central and highest redshift bins of
the PFS survey, respectively. To highlight the most significant lensing
impact, the power spectra shown here are the results with $k_{\parallel}=0$,
and the results multiplied by $k_{\perp}^{3/2}$ are plotted as function of the
transverse wavenumber, $k_{\perp}$. The red-dashed lines represent the lensing
contributions (i.e., $P_{EE}^{\rm(GL)}$ and $P_{\mu E}^{\rm(GL)}$ in upper and
lower panels, respectively), which are computed from equations (41) and (42),
adopting the top-hat survey window ($P_{EE}^{\rm(GL)}$). The redshift bin
size, $\Delta z=z_{\max}-z_{\min}$, is $\Delta z=0.2$ for $z=0.7$ and $1.3$,
and $\Delta z=0.4$ for $z=2.2$. The black-solid lines are the un-lensed power
spectra, $P_{EE}^{\rm(I)}$ (upper) and $P_{gE}^{\rm(I)}$ (lower), originated
purely from the IA and clustering. In upper panels, we also show the non-
vanishing noise contribution (see equation (30)), depicted as blue-dotted
lines.

Similarly, the observed density field is altered by gravitational lensing,
known as the magnification effect. By denoting the observed galaxy density
field as $\delta_{\rm obs}$, one can decompose it into the intrinsic density
and the term due to magnification, $\delta_{\rm obs}=\delta_{g}+\delta_{\mu}$.
Then the cross power spectrum between galaxy density and ellipticity fields,
$P_{gE}$, is expressed as

$\displaystyle P_{gE}(\bm{k};z)=P_{gE}^{\rm(I)}(\bm{k};z)+P_{\mu
E}^{\rm(GL)}(\bm{k};z),$ (40)

where the first term is the cross power spectrum between intrinsic density and
ellipticity fields considered so far, and the second term represents the lens-
induced cross-power spectrum. Again, there are also cross-talk terms, the
galaxy density–lensing shear $P_{gE}^{\rm(GL)}$ and magnification–intrinsic
ellipticity $P_{\mu E}^{\rm(I)}$ correlations. Furthermore, the lens-induced
ellipticities would be correlated with the kSZ, leading to a non-zero
contamination to $P_{vE}$. Since we consider the correlation functions in
individual $z$-slices, these cross-talks are negligible in our analysis.

Under the Limber approximation, $P_{EE}^{\rm(GL)}$ and $P_{\mu E}^{\rm(GL)}$
are analytically expressed as an integral of the comoving distance (e.g.
Matsubara, 2000; Hui et al., 2008):

$\displaystyle P_{EE}^{\rm(GL)}(\bm{k};\,z)=$
$\displaystyle\left(\frac{3}{2}\frac{\Omega_{\rm
m}H_{0}^{2}}{c^{2}}\right)^{2}|W_{\parallel}(k_{\parallel})|^{2}$
$\displaystyle\times\int_{0}^{\infty}d\chi^{\prime}\bigl{\\{}w\bigl{(}\chi^{\prime};\chi(z)\bigr{)}\bigr{\\}}^{2}\Bigl{\\{}\frac{\chi(z)}{\chi^{\prime}}\Bigr{\\}}^{2}$
$\displaystyle\ \ \ \ \ \ \ \ \times P_{\rm
lin}\left(\frac{\chi(z)}{\chi^{\prime}}k_{\perp};\,z(\chi^{\prime})\right),$
(41) $\displaystyle P_{\mu E}^{\rm(GL)}(\bm{k};z)=$ $\displaystyle
2(\alpha_{s}-1)P_{EE}^{\rm(GL)},$ (42)

where $\alpha_{s}$ is the logarithmic slope of the cumulative galaxy
luminosity function and the lensing kernel $w(\chi^{\prime};\chi)$ is given by
$w(\chi^{\prime};\chi)=(\chi-\chi^{\prime})\chi^{\prime}/\\{a(\chi^{\prime})\,\chi\\}\,\Theta(\chi-\chi^{\prime})$
with $\Theta(x)$ being the Heaviside step function. The function
$W_{\parallel}(k_{\parallel})$ is the Fourier counterpart of the survey window
function along the line-of-sight direction, $W_{\parallel}(x_{\parallel})$.
Equation (42) coincides with equation (15) of Ref. Hui et al. (2008), ignoring
the transverse survey window function $W_{\perp}$. Since our analysis targets
spectroscopic surveys with an accurate redshift determination provided for
each sample, we assume a top-hat window function,

$W_{\parallel}(x_{\parallel})=1/\sqrt{L},\ \ \ \text{if}\ \
\bar{\chi}-L/2<x_{\parallel}<\bar{\chi}+L/2,$ (43)

and $W_{\parallel}(x_{\parallel})=0$ otherwise. Here $L$ is the radial
comoving size which corresponds to the redshift bin, given by
$L=\chi(z_{\max})-\chi(z_{\min})\simeq(z_{\max}-z_{\min})\,c/H(z)$ (see table
2 for the values of $z_{\max}$ and $z_{\min}$ for each redshift bin). This
top-hat window leads to
$|W_{\parallel}(k_{\parallel})|^{2}=(4/Lk_{\parallel}^{2})\\{\sin{(k_{\parallel}L/2)}\\}^{2}$
in Fourier space. This means that the lensing contribution becomes maximum at
$k_{\parallel}\ll 1$, yielding $|W_{\parallel}|^{2}\sim L$.

Figure 12 shows $P_{EE}$ (upper row) and $P_{gE}$ (lower row) at $z=0.7$,
$1.30$, and $2.20$, which are the lowest, central and highest redshift bins of
the PFS survey, respectively. The power spectra shown here are the results
with $k_{\parallel}=0$ to highlight the maximum lensing contributions,
$P_{EE}^{\rm(GL)}$ (eq. 41) and $P_{\mu E}^{\rm(GL)}$ (eq. 42). As increasing
$z$, the amplitude of $P_{EE}^{\rm(GL)}$ depicted as red dashed lines in the
upper row, gets larger. However, apart from the shape noise, the signal coming
from the IA always dominates the E-mode power spectrum. Furthermore, the
amplitude of $P_{EE}^{\rm(GL)}$ is always smaller than the shape noise
expected from our fiducial setup of $\sigma_{\gamma}=0.2$, depicted as blue
dotted lines. On the other hand, for the power spectrum $P_{gE}$, the
amplitude is controlled by the additional parameter $\alpha_{s}$ (eq. (42)),
which depends on magnitude and redshift of a given galaxy sample. We adopt the
typical values of $\alpha_{s}$, $\alpha_{s}=2,2.5,$ and $3$ for $z=0.7$,
$1.30$, and $2.20$, respectively (e.g., Joachimi et al., 2011). Due to the
extra redshift dependence on $\alpha_{s}$, the lensing contribution to $P_{\mu
E}^{\rm(GL)}$ increases faster toward higher $z$ that to $P_{EE}^{\rm(GL)}$.
Nevertheless, the lensing contribution is still subdominant, and we can
clearly detect the BAO signal even for the case of $k_{\parallel}=0$.

Taking the lens-induced $E$-mode ellipticity and galaxy density fields to be
systematic errors, we have repeated the Fisher matrix analysis, for which the
lens-induced auto power spectrum of $E$-mode and cross power spectrum of
magnification and $E$-mode are included in the covariance at Eq. (27). We then
confirmed that the changes in the estimated errors are negligibly small.
Furthermore, instead of the top-hat filter function, we have examined another
filter, the Gaussian window function, given by
$|W_{\parallel}(k_{\parallel})|^{2}=\sqrt{4\pi\Sigma^{2}}\,\exp(-k_{\parallel}^{2}\Sigma^{2})$
in Fourier space. If we assume $\Sigma=L/\sqrt{4\pi}$, the contribution
becomes almost equivalent to the case with the top-hat window Hui et al.
(2008). If we choose a wider window, e.g., $\Sigma=L$, the amplitude of the
lens-induced power spectrum becomes $\sqrt{4\pi}\sim 3.5$ times larger. Even
in that case, changes in the statistical error on each parameter are still
negligible, $<1\%$, namely at most the last digits of the values quoted in
table 5 are modulated.

Hence, we conclude that the lensing effect on the observations of the IA gives
a sub-dominant contribution to $P_{EE}$ and $P_{gE}$ as long as we consider
spectroscopic surveys, and it hardly changes the cosmological constraints.

##  VI Conclusions

In this paper, based on the Fisher matrix analysis, we have shown that
combining IA and kSZ statistics with the conventional galaxy clustering
statistics substantially improves the geometric and dynamical constraints on
cosmology. As a representative of deep galaxy surveys for the forecast study,
we considered the Subaru PFS, whose angular area perfectly overlaps with those
from the HSC survey and the CMB-S4 experiment. We found that even without the
galaxy clustering, observations of IA and kSZ enable us to constrain $D_{\rm
A}$ and $H$, with the achievable precision down to a few percent. This
demonstrated that constraining the geometric distances with kSZ and IA effects
would help addressing recent systematics-related issues such as the Hubble
tension.

For cosmological parameter estimations, a relative merit of adding kSZ and IA
statistics to the galaxy clustering depends on cosmological models. We found
that the improvement of combining kSZ and IA to clustering statistics is
maximized if we simultaneously constrain the time-varying dark energy
equation-of-state parameter $w(a)=w_{0}+(1-a)w_{a}$ and the growth index
$\gamma$ characterizing the modification of gravity in a non-flat universe
($w_{0}w_{a}\gamma$ non-flat model). In such a model, with the CMB prior
information from the Planck experiment, the PFS-like deep survey is shown to
improve the constraints by $31\%$ for $\gamma$ and $>35\%$ for others except
the prior-dominated constraint on $\Omega_{K}$.

To see the gain of adding IA and kSZ for a different survey setup, we have
also performed the Fisher matrix analysis for the Euclid-like wide galaxy
survey, whose survey area is partly overlapped by half with the CMB-S4
experiment on the sky. Due to the large volume, such a wide survey can give
tighter constraints on $f$, $D_{\rm A}$ and $H$ at each redshift bin. However,
when considering the cosmological models which vary the growth index
parameter, a deep survey is more effective than a wide survey, and can get
tighter constraints. As a result, in the $w_{0}w_{a}\gamma$ non-flat model, by
combining kSZ and IA measurements with the clustering measurement,
cosmological constraints from the PFS-like deep survey can be tighter than
those with the conventional clustering-only measurement from the Euclid-like
wide survey. Finally, we have also discussed the potential impact of the
lensing effect on the observation of IA and line-of-sight structures on the
kSZ statistics, the former of which can systematically change the IA auto-
power spectrum, $P_{EE}$ (see equation (40)). However, even for the deep
survey considered, the lens-induced ellipticity is shown to give a negligible
contribution as long as we consider the three-dimensional power spectrum, and
hence the cosmological parameter estimated from the IA data is hardly changed.
For the kSZ statistics, even with a large correlation length of $D_{\rm
LOS}\sim 40-60\,h^{-1}$ Mpc, the impact of the line-of-sight structures on the
cosmological parameters is fairly small as long as we consider a joint
analysis with the galaxy clustering.

Table 6: Same as table 5 but one-dimensional fractional marginalized errors on cosmological parameters, $\sigma/\theta^{\rm fid}$, when only the data up to the linear scales $k_{\max}=0.1\,h\,{\rm Mpc}^{-1}$ are used.  |  |  | Deep survey |  | Wide survey
---|---|---|---|---|---
|  |  | Clustering |  |  |  |  | Clustering |  |  |
Model | $\sigma/\theta^{\rm fid}$ |  | only | +kSZ | +IA | +IA+kSZ |  | only | +kSZ | +IA | +IA+kSZ
$w_{0},w_{a}$ | $\Omega_{\rm m}$ |  | 0.163 | 0.142 | 0.138 | 0.128 |  | 0.117 | 0.111 | 0.106 | 0.105
flat | $w_{0}$ |  | 0.452 | 0.415 | 0.404 | 0.377 |  | 0.327 | 0.317 | 0.309 | 0.303
| $w_{a}$ |  | 1.162 | 1.047 | 1.021 | 0.952 |  | 0.857 | 0.833 | 0.813 | 0.795
| $H_{0}$ |  | 0.0785 | 0.0679 | 0.0657 | 0.0608 |  | 0.0538 | 0.0509 | 0.0484 | 0.0475
$w_{0},w_{a}$ | $\Omega_{\rm m}$ |  | 0.185 | 0.163 | 0.158 | 0.145 |  | 0.128 | 0.124 | 0.117 | 0.115
non-flat | $w_{0}$ |  | 0.518 | 0.465 | 0.452 | 0.415 |  | 0.371 | 0.358 | 0.344 | 0.337
| $w_{a}$ |  | 1.377 | 1.206 | 1.175 | 1.077 |  | 0.964 | 0.928 | 0.896 | 0.874
| $H_{0}$ |  | 0.0912 | 0.0805 | 0.0777 | 0.0715 |  | 0.0626 | 0.0607 | 0.0568 | 0.0560
| $100\Omega_{K}$ |  | 0.394 | 0.382 | 0.375 | 0.368 |  | 0.233 | 0.233 | 0.227 | 0.226
$w_{0},w_{a},\gamma$ | $\Omega_{\rm m}$ |  | 0.277 | 0.187 | 0.179 | 0.150 |  | 0.170 | 0.152 | 0.135 | 0.126
flat | $w_{0}$ |  | 0.786 | 0.557 | 0.535 | 0.447 |  | 0.485 | 0.446 | 0.400 | 0.368
| $w_{a}$ |  | 1.862 | 1.359 | 1.310 | 1.107 |  | 1.222 | 1.140 | 1.032 | 0.948
| $H_{0}$ |  | 0.1360 | 0.0910 | 0.0867 | 0.0721 |  | 0.0795 | 0.0708 | 0.0623 | 0.0580
| $\gamma$ |  | 0.499 | 0.375 | 0.370 | 0.326 |  | 0.349 | 0.323 | 0.300 | 0.259
$w_{0},w_{a},\gamma$ | $\Omega_{\rm m}$ |  | 0.345 | 0.217 | 0.207 | 0.169 |  | 0.243 | 0.208 | 0.168 | 0.151
non-flat | $w_{0}$ |  | 0.986 | 0.633 | 0.605 | 0.492 |  | 0.718 | 0.616 | 0.507 | 0.447
| $w_{a}$ |  | 2.410 | 1.583 | 1.521 | 1.248 |  | 1.792 | 1.547 | 1.291 | 1.133
| $H_{0}$ |  | 0.1722 | 0.1080 | 0.1027 | 0.0838 |  | 0.1212 | 0.1033 | 0.0834 | 0.0747
| $100\Omega_{K}$ |  | 0.439 | 0.392 | 0.384 | 0.371 |  | 0.305 | 0.285 | 0.259 | 0.247
| $\gamma$ |  | 0.559 | 0.386 | 0.380 | 0.329 |  | 0.459 | 0.397 | 0.343 | 0.284

In this paper, focusing specifically on the measurements of geometric and
dynamical distortions, we have shown that the combination of both IA and kSZ
with galaxy clustering is beneficial. Note, however, that the present analysis
using only the BAO and RSD information is not as powerful as that using the
full shape of the the underlying matter power spectrum. Although one advantage
in the present analysis is that the systematics arising from the nonlinearity
is less severe and hence conservative, it would be highly desirable for more
tighter cosmological constraints, in particular on the neutrino masses, to
make use of the full shape information Boyle and Komatsu (2018). Indeed, the
analysis with the full shape of the power spectrum has been performed in the
conventional galaxy clustering analysis Okumura et al. (2008); Ivanov et al.
(2020); d’Amico et al. (2020); Philcox et al. (2020); Kobayashi et al. (2022).
However, the analysis with full-shape information needs a proper nonlinear
modeling, and compared to the modeling of the nonlinearities for clustering
statistics, less studies have been made for velocity and ellipticity
statistics (e.g., Okumura et al., 2014; Blazek et al., 2019; Vlah et al.,
2020). Thus, before we extend our joint analysis of density, velocity and
ellipticity fields to include the full-shape spectra, we need to develop
models of nonlinear power spectra for the velocity and ellipticity fields and
test them with numerical simulations. Such analytical and numerical analyses
will be performed in future work.

###### Acknowledgements.

We would like to thank Masahiro Takada, Ryu Makiya, Eiichiro Komatsu, Naonori
Sugiyama and Sunao Sugiyama for helpful discussions. T. O. also thanks the
Subaru PFS Cosmology Working Group for useful correspondences during the
regular telecon. T. O. acknowledges support from the Ministry of Science and
Technology of Taiwan under Grants Nos. MOST 110-2112-M-001-045- and
111-2112-M-001-061- and the Career Development Award, Academia Sinina (AS-
CDA-108-M02) for the period of 2019 to 2023. A. T. was supported by MEXT/JSPS
KAKENHI Grant Number JP17H06359, JP20H05861 and JP21H01081. A. T. also
acknowledges financial support from Japan Science and Technology Agency (JST)
AIP Acceleration Research Grant Number JP20317829.

##  Appendix A CMB prior

In this Appendix, we describe the CMB prior information added to the Fisher
matrix of the LSS probes (see figure 1). In the analysis presented in this
paper, the CMB prior information is used to estimate the forecast constraints
on cosmological parameters, except for the minimal cosmological model ($w_{0}$
flat model).

First of all, our primary interest is how the geometric and dynamical
constraints derived from the BAO and RSD measurements can be used to test
cosmological models, with the power spectra of each LSS probe characterized by
Eq. (37). For this purpose, we specifically use the information determined
mainly from the CMB acoustic scales. We follow Ref. Aubourg et al. (2015) and
use the information on $\omega_{\rm cb}\equiv\Omega_{\rm cb}h^{2}$ and $D_{\rm
M}(1090)/r_{\rm d}$, fixing the energy density of neutrinos $\omega_{\nu}$ and
baryon $\omega_{\rm b}$ respectively to $\omega_{\nu}=6.42\times 10^{-4}$ and
$\omega_{\rm b}=0.022284$, the former of which corresponds to the total mass
of $\sum{m_{\nu}}=0.06$ [eV]. Here, the $\Omega_{\rm cb}$ is the density
parameter of CDM and baryons, i.e., $\Omega_{\rm cb}=\Omega_{\rm
c}+\Omega_{\rm b}$. The quantity $D_{\rm M}(z)=(1+z)D_{\rm A}(z)$ is the
comoving angular diameter distance Hogg (1999), and $r_{d}$ is the sound
horizon at the drag epoch, for which we use the numerically calibrated
approximation:

$r_{\rm
d}\simeq\frac{55.124\exp{\left[-72.3(\omega_{\nu}+0.0006)^{2}\right]}}{\omega_{\rm
cb}^{0.25351}\omega_{\rm b}^{0.12807}}~{}{\rm Mpc},$ (44)

with $\omega_{\nu}$ and $\omega_{\rm b}$ kept fixed to the values mentioned
above. Ref. Aubourg et al. (2015) found that the acoustic scale information on
the data vector $\Theta_{\alpha}=(\omega_{\rm cb},D_{\rm M}(1090)/r_{\rm d})$
can be described by a Gaussian likelihood with mean and covariance (see also
Ivanov et al. (2020))333To be precise, Ref. Aubourg et al. (2015) provided a
Gaussian likelihood for the three parameters $\Theta_{\alpha}=(\omega_{\rm
b},\,\omega_{\rm cb},\,D_{\rm M}(1090)/r_{\rm d}$) having the $3\times 3$
covariance matrix. Since we consider $\omega_{\rm b}$ to be fixed, the
relevant prior information is described by the $2\times 2$ covariance matrix
given at equation (46).:

$\displaystyle\mu_{\Theta}=\left(0.1386,\,94.33\right),$ (45) $\displaystyle
C_{\Theta}=\begin{pmatrix}7.452\times 10^{-6}&-3.605\times 10^{-5}\\\
-3.605\times 10^{-5}&0.004264\end{pmatrix}.$ (46)

The inverse of this error matrix is the Fisher matrix, $\bm{F}_{{\rm
CMB}}=\bm{C}_{\Theta}^{-1}$, shown in the lower left of the flowchart in
figure 1. It is then converted to the Fisher matrix for a given cosmological
model of interest, $\bm{S}_{{\rm CMB}}$, through equation (33). We have also
tried another CMB prior used in our early study Taruya and Okumura (2020),
based on Seo & Eisenstein Seo and Eisenstein (2003), and confirmed that our
forecast results almost remain unchanged.

##  Appendix B Forecast results with the conservative cutoff of
$k_{\max}=0.1h{\rm Mpc}^{-1}$

Figure 13:  Cosmological constraints on the $w_{0}w_{a}\gamma$ non-flat model
from the deep (Top ) and wide (Bottom) surveys. These results are similar with
figure 8 but using the conservative range of data of $k_{\max}=0.1\,h\,{\rm
Mpc}^{-1}$.

In sections IV and V, forecast constraints on cosmological parameters,
including the geometric distances and growth of structure, were derived
focusing on the upcoming deep and wide galaxy surveys, PFS and Euclid,
respectively. In doing so, one important assumption was that the linear theory
template for the power spectra is applicable to the weakly nonlinear scales,
setting the maximum wavenumber to $k_{\rm max}=0.2\,h$ Mpc-1 for all the three
LSS probes. While our analysis is still conservative in the sense that we only
use the geometric and dynamical information obtained from the BAO and RSD
measurements, restricting the data to the linear scales of $k\leq 0.1\,h\,{\rm
Mpc}^{-1}$ would yield a more conservative and robust forecast results, and no
intricate modeling of the nonlinear systematics needs to be developed. In this
appendix, repeating the Fisher matrix analysis but with $k_{\rm max}=0.1\,h$
Mpc-1, we summarize the forecast constraints on cosmological parameters.

First we consider the deep survey. The left half of table 6 summarizes the
one-dimensional marginalized errors on cosmological parameters, which are
compared to results with $k_{\rm max}=0.2\,h$ Mpc-1 listed in the left half of
table 5. The results are also shown visually as the hollow bars in figure 9.
The expected errors obtained from the clustering-only analysis with $k_{\rm
max}=0.1\,h$ Mpc-1 are roughly twice as large as those with $k_{\rm
max}=0.2\,h$ Mpc-1. Interestingly, however, the fractional gain of the
cosmological power by adding the kSZ and/or IA measurements is more
significant for the conservative analysis with $k_{\max}=0.1\,h\,{\rm
Mpc}^{-1}$. For instance, in the most general model considered in this paper,
namely the $w_{0}w_{a}\gamma$ non-flat model (see table 4), the improvements
by $48\%$ and $41\%$, relative to the clustering-only analysis are
respectively achieved for the constraints on $w_{a}$ and $\gamma$. These are
compared to the relative improvements by $35\%$ and $31\%$ in the cases with
$k_{\max}=0.2\,h\,{\rm Mpc}^{-1}$.

Let us then compare the forecast results for the deep survey with those for
the wide galaxy survey. As seen in the right side of table 6 (see also the
hollow bars in figure 9), the constraining power of the clustering-only
analysis from the wide survey is $25-40\%$ stronger than that from the deep
survey. This is more or less the same as the case with the aggressive cutoff
of $k_{\max}=0.2\,h\,{\rm Mpc}^{-1}$. However, one notable point is that the
benefit of combining the IA and kSZ measurements is more significant for the
deep survey than that for the wide survey. In particular, in the
$w_{0}w_{a}\gamma$ non-flat model, combining either IA or kSZ with clustering
in the deep survey can beat the constraining power of the wide survey. For
illustration, in figure 13, the expected two-dimensional error contours on the
cosmological parameters are shown in the $w_{0}w_{a}\gamma$ non-flat model.
This figure is similar with figure 8, but here we adopt the conservative cut,
$k_{\max}=0.1\,h\,{\rm Mpc}^{-1}$, instead of $0.2\,h\,{\rm Mpc}^{-1}$.
Clearly, the relative impact of combining IA and kSZ is rather large for the
deep survey, manifesting tighter constraints not only on the growth index but
also on other parameters including the curvature parameter.

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