# Full-shape cosmology analysis of SDSS-III BOSS galaxy power spectrum using
emulator-based halo model: a $5\%$ determination of $\sigma_{8}$

Yosuke Kobayashi1,2<EMAIL_ADDRESS>Takahiro Nishimichi3,2
Masahiro Takada2<EMAIL_ADDRESS>Hironao Miyatake4,2 1 Department of
Astronomy/Steward Observatory, University of Arizona, 933 North Cherry Avenue,
Tucson, AZ 85721-0065, USA
2 Kavli Institute for the Physics and Mathematics of the Universe (WPI), The
University of Tokyo Institutes for Advanced Study (UTIAS), The University of
Tokyo, Chiba 277-8583, Japan
3 Center for Gravitational Physics, Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, Japan
4 Kobayashi-Maskawa Institute for the Origin of Particles and the Universe
(KMI), Nagoya University, Nagoya, 464-8602, Japan

###### Abstract

We present the results obtained from the full-shape cosmology analysis of the
redshift-space power spectra for 4 galaxy samples of the SDSS-III BOSS DR12
galaxy catalog over $0.2<z<0.75$. For the theoretical template, we use an
emulator that was built from an ensemble set of $N$-body simulations, which
enables fast and accurate computation of the redshift-space power spectrum of
“halos”. Combining with the halo occupation distribution to model the galaxy-
halo connection, we can compute the redshift-space power spectrum of BOSS-like
galaxies in less than a CPU second, for an input model under flat $\Lambda$CDM
cosmology. In our cosmology inference, we use the monopole, quadrupole and
hexadecapole moments of the redshift-space power spectrum and include 7
nuisance parameters, with broad priors, to model uncertainties in the galaxy-
halo connection for each galaxy sample, but do not use any information on the
abundance of galaxies. We demonstrate a validation of our analysis pipeline
using the mock catalogs of BOSS-like galaxies, generated using different
recipes of the galaxy-halo connection and including the assembly bias effect.
Assuming weak priors on cosmological parameters, except for the BBN prior on
$\Omega_{\mathrm{b}}h^{2}$ and the CMB prior on $n_{\mathrm{s}}$, we show that
our model well reproduces the BOSS power spectra. Including the power spectrum
information up to $k_{\mathrm{max}}=0.25\,h\,\mathrm{Mpc}^{-1}$, we find
$\Omega_{\mathrm{m}}=0.301^{+0.012}_{-0.011}$, $H_{0}=68.2\pm
1.4~{}\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$, and
$\sigma_{8}=0.786^{+0.036}_{-0.037}$, for the mode and 68% credible interval,
after marginalization over galaxy-halo connection parameters. We find little
improvement in the cosmological parameters beyond a maximum wavelength
$k_{\mathrm{max}}\simeq 0.2\,h\,\mathrm{Mpc}^{-1}$ due to the shot noise
domination and marginalization of the galaxy-halo connection parameters. Our
results are consistent with the Planck CMB results within $1\sigma$
statistical uncertainties.

††preprint: IPMU21-0058††preprint: YITP-21-112

##  I Introduction

The three-dimensional distribution of galaxies, measured from wide-area
spectroscopic surveys of galaxies, is a powerful probe of cosmology, e.g., for
constraining cosmological parameters such as parameters characterizing the
nature of dark energy and for testing gravity theory on cosmological scales
[1, 2, 3, 4, 5, 6, 7]. To attain the fundamental cosmology, there are various
exiting, ongoing and planned galaxy redshift surveys: the SDSS-III Baryon
Oscillation Spectroscopic Survey (BOSS; [8]), the SDSS-IV extended Baryon
Oscillation Spectroscopic Survey (eBOSS; [9]), the Subaru Prime Focus
Spectrograph (PFS; [10]), the Dark Energy Spectroscopic Instrument (DESI;
[11]), the ESA Euclid satellite mission [12], and the NASA Nancy Grace Roman
Space Telescope [13].

The galaxy distribution observed by spectroscopic surveys is modulated by the
Doppler effect due to the line-of-sight peculiar velocities of galaxies, and
exhibits characteristic anisotropies, called the redshift-space distortion
(RSD) [14, 15, 16]. The RSD effect is useful to improve cosmological
constraints by breaking degeneracies between the cosmological parameters and
uncertainties in galaxy bias relative to the underlying matter distribution
[17]. In addition, since the RSD effect is a gravitational effect, it can be
used, if precisely measured, to probe the strength of gravitational field in
large-scale structure, which can be in turn used to test gravity theory on
cosmological scales [18].

In order to exploit the full information from galaxy redshift surveys, we need
a sufficiently accurate theoretical template that enables a high-fidelity
comparison with the measured clustering statistics of galaxies to obtain a
robust estimation of cosmological parameters. The linear theory of
cosmological fluctuations, which has been in a remarkable success in CMB
analyses, ceases to be accurate at $k\gtrsim 0.1\,h{\rm Mpc}^{-1}$ due to
nonlinear effects of structure formation [17]. The standard approach to tackle
this difficulty has been analytic prescriptions based on the perturbation
theory (PT) of large-scale structure [19, 20]. This approach describes the
distribution of galaxies in terms of a series expansion of both the matter
density and velocity fields with a set of free coefficients/terms including
bias parameters, under the single-stream approximation [21, 22]. A further
refined model consistently separating short-scale physics including the galaxy
bias from large-scale dynamics of interest, so-called Effective Field Theory
of Large-Scale Structure (EFTofLSS), has also been developed [23]. These
models have been applied to actual datasets to obtain cosmological constraints
[24, 25, 26, 5, 27, 28, 29]. While these PT-based templates give useful
predictions at linear and quasi-nonlinear scales up to $k\sim
0.2\,h\,\mathrm{Mpc}^{-1}$, an application of these models to even smaller
scales is still disturbed by even higher-order contributions of both the
density and velocity fields as well as non-perturbative effects arising from
the dynamics beyond shell crossing, i.e., formation of galaxies (or dark
matter halos) [e.g., 30, 31, 32, 33, 34, 35, 36]. Consequently, the cosmology
analysis of the galaxy power spectrum has been typically limited to the
wavenumber $k\lesssim 0.15$ – $0.2\,h\,\mathrm{Mpc}^{-1}$, depending on the
redshift and the accuracy of the model required to meet the statistical
precision of data [26, 25, 37]. In other words, the clustering information on
the higher-$k$ scales does not seem practical for cosmology in this method,
because the information is used to basically constrain higher-order bias
parameters and other nuisance parameters that need to be introduced for the
theoretical consistency of models.

As an alternative approach, in this paper we use a simulation-based
theoretical template, the emulator, which enables fast and accurate
computation of the redshift-space power spectrum of “halos” in the flat
$\Lambda$CDM framework, developed in our previous paper [38]. Dark matter
halos are locations where galaxies likely form. It is relatively
straightforward to accurately simulate the formation and evolution processes
of halos using $N$-body simulations and then have an accurate prediction of
their clustering properties including the redshift-space power spectrum [39,
20]. Kobayashi et al. [38] developed an emulator by training a feed-forward
neural network with a dataset of the redshift-space power spectra of halos
measured from halo catalogs in an ensemble set of $N$-body simulations for 80
models within flat $w$CDM framework in the Dark Quest campaign [40]. The
emulator was validated using the test dataset consisting of 20 cosmological
models that are not in training, and it was shown that the power spectra are
sufficiently accurate up to $k=0.6\,h\,\mathrm{Mpc}^{-1}$. The emulator
includes all the non-perturbative, nonlinear effects relevant to the formation
and evolution of halos: nonlinear clustering, nonlinear RSD, nonlinear bias of
halos, and halo exclusion effect. By combining with a halo occupation
distribution (HOD) model [41, 42, 43, 44, 45, 46], the emulator enables to
compute the redshift-space power spectrum of galaxies in less than a CPU
second on a typical recent laptop computer. Thus, the emulator allows for
cosmological parameter inference of the galaxy power spectrum in a multi-
dimensional parameter space. Such an emulator-based method is equivalent to
estimating cosmological parameters from comparison of the observed power
spectrum with mock spectra from simulated galaxy catalogs generated from
costly high-resolution $N$-body simulations with varying cosmological models.

Hence, the purpose of this paper is to perform a cosmology analysis of the
redshift-space galaxy power spectrum measured from the public BOSS DR12 large-
scale structure catalog over $0.2<z<0.75$
111https://data.sdss.org/sas/dr12/boss/lss/, using the aforementioned
emulator. In doing this, we analyze the “full-shape” information in the
monopole, quadrupole and hexadecapole moments of the redshift-space power
spectra, beyond the traditional approach to extract only geometrical
information through the features of baryon acoustic oscillations (BAO) and the
anisotropy originating from RSD. We first show a series of validation checks
of our method to ensure unbiased cosmological inference beyond the accuracy
assessment of the emulator at the halo power spectrum level already presented
in our previous paper. We apply the full analysis pipeline to mock signals of
the galaxy power spectrum measured from the galaxy mock catalogs and then
check whether our method can recover the cosmological parameters to within the
statistical errors. For this validation, we use the mock catalogs generated
using different recipes of galaxy-halo connection from our fiducial HOD model
[also see 37, for similar analyses but with different observables] and also
use mock catalogs including the assembly bias effect that is one of the most
dangerous, physical systematic effects in the halo model approach. We then
apply the analysis method to the BOSS galaxy power spectra under the flat
$\Lambda$CDM cosmology. In doing this, we employ weak priors on the
cosmological parameters, except for the BBN prior on
$\Omega_{\mathrm{b}}h^{2}$ [48] and the CMB prior on $n_{\mathrm{s}}$ [49],
employ very broad priors for the galaxy-halo connection parameters, and then
estimate the cosmological parameters after marginalization over uncertainties
in the nuisance parameters. We also compare our cosmological constraints with
the recent PT-based results [50, 51] and the Planck 2018 cosmological results
[49].

The structure of this paper is as follows. In Sec. II we will describe the
power spectrum data and covariance we use in this work. In Sec. III we will
describe details of the emulator-based halo model as the theoretical template
and then the parameters and priors used in the cosmology analysis. In Sec. IV
we show the main results of this paper: the cosmological parameters for the
flat $\Lambda$CDM model. In Sec. V we give discussion on possible residual
systematic effects in the data and the theoretical template. Sec. VI is
devoted to the conclusion.

##  II Data

For the galaxy power spectrum data and the covariance matrix, we use the
updated measurement of the BOSS DR12 power spectrum recently provided in Ref.
[52], which are publicly available from
https://fbeutler.github.io/hub/deconv_paper.html. We use four data chunks,
named “NGC $z1$”, “SGC $z1$”, “NGC $z3$”, and “SGC $z3$”, which mean the
measurements for different galaxy samples in two non-overlapping redshift bins
(“$z1$” and “$z3$”) both for the Northern and Southern Galactic caps (“NGC”
and “SGC”). “$z1$” (“$z3$”), which we hereafter call “low-$z$” (“high-$z$”),
corresponds to the range $0.2<z<0.5$ ($0.5<z<0.75$) with its effective
redshift $z_{\mathrm{eff}}=0.38$ (0.61). We compute the theoretical model of
the power spectrum for each redshift bin at its effective redshift.

The estimator adopted in Ref. [52] gives the power-spectrum multipoles
convolved with the survey window function, which can be expressed by a matrix-
vector multiplication. Therefore, we construct a consistent model prediction
of the window-convolved power spectrum as

$\displaystyle\mathbf{P}^{\mathrm{conv}}=\mathbf{W}\mathbf{M}\mathbf{P}^{\mathrm{true,flat-
sky}},$ (1)

where $\mathbf{P}^{\mathrm{conv}}$ is the vector of the convolved power
spectrum multipoles, $\mathbf{P}^{\mathrm{true,flat-
sky}}=\\{P_{0},P_{2},P_{4}\\}$ is that of the true (unconvolved) power
spectrum multipoles computed in the global plane-parallel approximation, where
the monopole, quadrupole, and hexadecapole moments are concatenated.
$\mathbf{M}$ is the matrix that induces the odd multipoles (dipole and
octopole) arising from the wide-angle effect. $\mathbf{W}$ is the matrix that
consists of Fourier-space multipole moments of the survey window function (see
Ref. [52] for details). Due to the correction for the wide-angle effect
through the matrix $\mathbf{M}$, the dipole and octopole moments (the real
parts, since they are purely imaginary) are induced by the leakage from the
standard even multipole moments, but we do not use these odd multipoles in
this work. We compute the power spectrum monopole, quadrupole, and
hexadecapole using the emulator-based theoretical model we describe in the
next section and substitute them to $\mathbf{P}^{\mathrm{true,flat-sky}}$ in
Eq. (1). The matrices, $\mathbf{W}$ and $\mathbf{M}$, are also made publicly
available on the same website.

In the measurement of the convolved power spectrum multipoles, Ref. [52]
assumes the flat-geometry $\Lambda$CDM model with $\Omega_{\mathrm{m}}=0.31$,
which is the same as the model assumed in the original measurement of the BOSS
DR12 public spectra for their cosmology analysis [26]. They measure the power
spectrum in a cubic box by a fast Fourier transform (FFT) with the side length
$L=3500\,h^{-1}\,\mathrm{Mpc}$ and Nyquist wavenumber
$k_{\mathrm{Ny}}=0.628\,h\,\mathrm{Mpc}^{-1}$. Each Fourier mode is divided
into wavenumber bins in the range $k\in(0,0.4)\,h\,\mathrm{Mpc}^{-1}$ with
width $\Delta k=0.01\,h\,\mathrm{Mpc}^{-1}$.

As for the power spectrum covariance matrix, we use the matrix provided by
Ref. [52], which was estimated from 2048 realizations of the MultiDark-Patchy
(hereafter Patchy) mock catalogs [53, 54]. These mock catalogs were generated
using an approximate $N$-body solver which combines the Lagrangian
perturbation theory, a small-scale halo collapse model, and a semi-analytical
galaxy biasing scheme, augmented by calibration to a reference large-volume
$N$-body simulation sample selected from the BigMultiDark simulations [55]. To
correct the biased estimate of the inverse covariance, we multiply the so-
called Hartlap factor [56] to the inverse of the estimated covariance:

$\displaystyle
f_{\mathrm{Hartlap}}=\frac{N_{\mathrm{r}}-n_{\mathrm{bin}}-2}{N_{\mathrm{r}}-1},$
(2)

where $n_{\mathrm{bin}}$ is the number of bins we use in the parameter
inference. For instance, in the case that we use the monopole, quadrupole and
hexadecapole moments up to $k_{\mathrm{max}}=0.25\,h\,\mathrm{Mpc}^{-1}$, the
number of bins is $n_{\mathrm{bin}}=25\times 3=75$ for each galaxy sample,
yielding $f_{\mathrm{Hartlap}}=0.9629$.

##  III Theoretical model and parameters

###  III.1 Theoretical model

To investigate the cosmological parameter constraint from the BOSS galaxy
power spectrum, we use the theoretical template computed using the emulator
for the redshift-space power spectrum of halos combined with the HOD model,
developed in our previous work [38]. We below give a brief description of our
theoretical template, and see Ref. [38] [also 17] for further details.

We employ the five-parameter HOD model in Ref. [45], which splits the galaxies
into central and satellite galaxies. The mean halo occupation number of
central and satellite galaxies within host halos with mass $M$ are given as

$\displaystyle\left\langle
N_{\mathrm{c}}\right\rangle\\!(M)=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\log
M-\log M_{\mathrm{min}}}{\sigma_{\log M}}\right)\right],$ (3)

and

$\displaystyle\left\langle N_{\mathrm{s}}\right\rangle\\!(M)$
$\displaystyle=\left\langle
N_{\mathrm{c}}\right\rangle\\!(M)\lambda_{\mathrm{s}}(M)$
$\displaystyle\equiv\left\langle
N_{\mathrm{c}}\right\rangle\\!(M)\left(\frac{M-\kappa
M_{\mathrm{min}}}{M_{1}}\right)^{\alpha_{\mathrm{sat}}},$ (4)

respectively, where $\mathrm{erf}(x)$ is the error function and the logarithms
in Eq. (3) are base 10. Note that, in our model we adopt $M\equiv
M_{\mathrm{200}}=(4\pi/3)200\bar{\rho}_{\mathrm{m0}}R_{200}^{3}$ as the halo
mass definition, where $\bar{\rho}_{\mathrm{m0}}$ is the mean comoving mass
density in the universe and $R_{200}$ is the spherical comoving radius within
which the mean mass density is 200 times $\bar{\rho}_{\mathrm{m0}}$. Here
$\\{\log M_{\mathrm{min}},\sigma^{2}_{\log M},\log
M_{1},\alpha_{\mathrm{sat}},\kappa\\}$ are model parameters. The probability
distribution of galaxies given the mean number is assumed to be Bernoulli for
centrals and Poisson for satellites given that the halo of interest has a
central halo. This HOD model is also used in the cosmology analysis of the
galaxy-galaxy weak lensing and projected galaxy correlation function measured
from the Hyper Suprime-Cam Year1 catalog and the BOSS DR11 catalog [57].

Using the above HOD model, our full model of the redshift-space galaxy power
spectrum is given by the sum of the one- and two-halo terms,

$\displaystyle P_{\mathrm{gg}}({\bf k})=P^{\mathrm{1h}}_{\mathrm{gg}}({\bf
k})+P^{\mathrm{2h}}_{\mathrm{gg}}({\bf k}),$ (5)

where

$\displaystyle P^{\mathrm{1h}}_{\mathrm{gg}}({\bf k})$
$\displaystyle=\frac{1}{\bar{n}_{\mathrm{g}}^{2}}\int\\!\\!\mathrm{d}M~{}{\frac{\mathrm{d}n}{\mathrm{d}M}(M)}\left\langle
N_{\mathrm{c}}\right\rangle\\!(M)$ $\displaystyle\hskip
10.00002pt\times\left[2\lambda_{\mathrm{s}}(M)\,\tilde{{\cal H}}({\bf
k};M)+\lambda_{\mathrm{s}}(M)^{2}\,\tilde{{\cal H}}({\bf k};M)^{2}\right],$
(6)

and

$\displaystyle P^{\mathrm{2h}}_{\mathrm{gg}}({\bf k})$
$\displaystyle=\frac{1}{\bar{n}_{\mathrm{g}}^{2}}\int\\!\mathrm{d}M_{1}{\frac{\mathrm{d}n}{\mathrm{d}M}(M_{1})}\left[\left\langle
N_{\mathrm{c}}\right\rangle\\!(M_{1})+\left\langle
N_{\mathrm{s}}\right\rangle\\!(M_{1})\,\tilde{{\cal H}}({\bf k};M_{1})\right]$
$\displaystyle\hskip
10.00002pt\times\int\\!\mathrm{d}M_{2}{\frac{\mathrm{d}n}{\mathrm{d}M}(M_{2})}\left[\left\langle
N_{\mathrm{c}}\right\rangle\\!(M_{2})+\left\langle
N_{\mathrm{s}}\right\rangle\\!(M_{2})\,\tilde{{\cal H}}({\bf k};M_{2})\right]$
$\displaystyle\hskip 10.00002pt\times{P_{\mathrm{hh}}({\bf k};M_{1},M_{2})},$
(7)

where $P_{\mathrm{hh}}({\bf k};M_{1},M_{2})$ is the redshift-space halo power
spectrum, for which we use our emulator [38] that includes the RSD effects in
both linear and nonlinear regimes for an input set of model parameters
(cosmological parameters, redshift and halo masses). These power spectra
depend on the two-dimensional wavevector, ${\bf k}=(k_{\parallel},k_{\perp})$
or $k(\mu,\sqrt{1-\mu^{2}})$, due to the redshift-space distortion effect,
where $\mu$ is the cosine angle between the wavevector ${\bf k}$ and the line-
of-sight direction $\hat{{\bf n}}$. In the above formulae,
$\mathrm{d}n/\mathrm{d}M$ is the halo mass function and $\bar{n}_{\mathrm{g}}$
is the global mean number density of galaxies defined as

$\displaystyle\bar{n}_{\mathrm{g}}=\int\\!\\!\mathrm{d}M~{}{\frac{\mathrm{d}n}{\mathrm{d}M}(M)}\left[\left\langle
N_{\mathrm{c}}\right\rangle\\!(M)+\left\langle
N_{\mathrm{s}}\right\rangle\\!(M)\right].$ (8)

For satellite galaxies, we model the intra-halo profile in redshift space as
the multiplication of the Navarro-Frenk-White (NFW) profile [58] and the
velocity distribution [59, 60, 17]:

$\displaystyle\tilde{\mathcal{H}}({\bf
k};M)=\tilde{u}_{\mathrm{NFW}}(k;M)\,\tilde{\mathcal{F}}(k_{\parallel};M),$
(9)

where $\tilde{u}_{\mathrm{NFW}}(k;M)$ is the Fourier transform of the NFW
density profile normalized by halo mass $M$. To specify the NFW profile, we
assume the median concentration-mass relation $c(M_{\mathrm{200}})$ following
the model in Refs. [61, 62]. $\tilde{\mathcal{F}}(k_{\parallel};M)$ is the
Fourier transform of the Gaussian velocity distribution whose velocity
dispersion is given as

$\displaystyle\mathcal{F}(\Delta
r_{\parallel};M)=\frac{1}{\sqrt{2\pi\sigma_{\mathrm{v}}^{2}(M)}}\exp\left[-\frac{\Delta
r_{\parallel}^{2}}{2\sigma_{\mathrm{v}}^{2}(M)}\right].$ (10)

The velocity dispersion, $\sigma_{\rm v}^{2}(M)$, is given by

$\displaystyle\sigma_{\mathrm{v}}^{2}(M)=\frac{1}{a^{2}H^{2}}\frac{GM}{2aR_{\mathrm{200}}(M)},$
(11)

where $G$ is the gravitational constant, $a$ is the scale factor, $H$ is the
Hubble parameter at scale factor $a$, and $R_{\mathrm{200}}(M)$ is the halo
radius $R_{\mathrm{200}}$ for halos with mass $M$. The factor $1/(aH)^{2}$ is
to convert the velocity to the redshift-space displacement. The above velocity
distribution models the fingers-of-God (FoG) effect due to virial motions of
satellite galaxies in their host halos. In order to further include
uncertainties in the FoG effect, we will introduce a nuisance parameter to
model the uncertainty, $\sigma_{{\rm v}}(M)\rightarrow c_{\rm vel}\sigma_{{\rm
v}}(M)$, and treat $c_{\rm vel}$ as a model parameter in the cosmology
analysis.

As described above, we use our emulator [38] to compute the redshift-space
halo power spectrum $P_{\mathrm{hh}}({\bf k};M_{1},M_{2})$ in the two-halo
term (Eq. III.1) for an input model in the flat $\Lambda$CDM cosmology. The
emulator outputs $P_{\rm hh}({\bf k};M_{1},M_{2})$ given as a function of the
two-dimensional wavevector ${\bf k}$, rather than the multipole moments such
as $P_{{\rm hh},\ell}(k;M_{1},M_{2})$. This modeling allows us to
straightforwardly include the Alcock-Paczyński effect (see the next
subsection), even in the presence of the FoG effect. We also use the publicly
available Dark Emulator (https://dark-emulator.readthedocs.io/en/latest/) [40]
to compute the halo mass function, and the Colossus code
(http://www.benediktdiemer.com/code/colossus/) [63] to compute the halo
concentration-mass relation. In the theoretical model the redshift-space halo
power spectrum in the two-halo term, $P_{\mathrm{hh}}({\bf k};M_{1},M_{2})$,
carries cosmological information, and other functions such as the HOD and the
distribution of galaxies in their host halos are all nuisance and are needed
to account for uncertainties in the galaxy-halo connection.

###  III.2 Alcock-Paczyński effect

Since we need to assume a reference cosmology ($\Omega_{\mathrm{m}}=0.31$) in
the measurement of the power spectrum, we incorporate the Alcock-Paczyński
(AP) effect [64, 65] into the theoretical model. Due to the difference between
the true and reference cosmologies, the Fourier-space wavenumbers are
transformed as

$\displaystyle k^{\mathrm{ref}}_{\perp}$
$\displaystyle=\alpha_{\perp}k_{\perp}\equiv\frac{D_{\mathrm{A}}(z)}{D^{\mathrm{ref}}_{\mathrm{A}}(z)}k_{\perp},$
$\displaystyle k^{\mathrm{ref}}_{\parallel}$
$\displaystyle=\alpha_{\parallel}k_{\parallel}\equiv\frac{H^{\mathrm{ref}}(z)}{H(z)}k_{\parallel},$
(12)

where $D_{\mathrm{A}}(z)$ and $H(z)$ are the angular diameter distance and
Hubble parameter at the effective redshift $z$. Quantities with superscript
“ref” denote those for the reference cosmology. By this coordinate
transformation, the power spectrum multipole is transformed as

$\displaystyle
P^{\mathrm{ref}}_{\mathrm{gg},\ell}(k^{\mathrm{ref}})=\frac{2\ell+1}{2\alpha^{2}_{\perp}\alpha_{\parallel}}\int_{-1}^{1}\\!\mathrm{d}\mu^{\mathrm{ref}}~{}P_{\mathrm{gg}}\\!(k,\mu)\mathcal{L}_{\ell}(\mu^{\mathrm{ref}}).$
(13)

where $k$ and $\mu$ in the argument of $P^{\mathrm{ref}}_{\mathrm{gg},\ell}$
are given in terms of $k^{\mathrm{ref}}$ and $\mu^{\mathrm{ref}}$;
$k=k(k^{\mathrm{ref}},\mu^{\mathrm{ref}})$ and $\mu=\mu(\mu^{\mathrm{ref}})$,
given as

$\displaystyle k(k^{\mathrm{ref}},\mu^{\mathrm{ref}})$
$\displaystyle=k^{\mathrm{ref}}\frac{1}{\alpha_{\perp}}\left[1+(\mu^{\mathrm{ref}})^{2}\left(\frac{\alpha_{\perp}^{2}}{\alpha_{\parallel}^{2}}-1\right)\right]^{1/2},$
(14) $\displaystyle\mu(\mu^{\mathrm{ref}})$
$\displaystyle=\mu^{\mathrm{ref}}\frac{\alpha_{\perp}}{\alpha_{\parallel}}\left[1+(\mu^{\mathrm{ref}})^{2}\left(\frac{\alpha_{\perp}^{2}}{\alpha_{\parallel}^{2}}-1\right)\right]^{-1/2}.$
(15)

Since the reference cosmology is generally different from the underlying true
cosmology, the AP effect induces an additional anisotropy in the measured
power spectrum. In other words, the measured anisotropy enables us to infer
the true cosmology through the geometrical information.

###  III.3 Model parameters and priors

In this subsection, we describe the model parameters that we infer, as well as
their prior settings. The default setting is summarized in Table 1.

Parameter | Prior
---|---
Cosmological parameters
$\omega_{\mathrm{b}}$ | $\mathcal{N}(0.02268,0.00038)$
$\omega_{\mathrm{c}}$ | $\mathcal{U}(0.10782,0.13178)$
$\Omega_{\Lambda}$ | $\mathcal{U}(0.54752,0.82128)$
$\ln(10^{10}A_{\mathrm{s}})$ | $\mathcal{U}(2.4752,3.7128)$
$n_{\mathrm{s}}$ | $\mathcal{N}(0.9649,0.0042)$
HOD parameters
$\log M_{\mathrm{min}}$ | $\mathcal{U}(12.0,15.0)$
$\sigma_{\log M}^{2}$ | $\mathcal{U}(0.0001,2.0)$
$\log M_{1}$ | $\mathcal{U}(12.0,16.0)$
$\alpha_{\mathrm{sat}}$ | $\mathcal{U}(0.01,5.0)$
$\kappa$ | $\mathcal{U}(0.01,5.0)$
Other nuisance parameters
$c_{\mathrm{vel}}$ | $\mathcal{U}(0.01,10.0)$
$P_{\mathrm{shot}}$ | $\mathcal{U}(-10^{4},10^{4})\,h^{-3}\,\mathrm{Mpc}^{3}$
Derived parameters
$\Omega_{\mathrm{m}}$ | $-$
$H_{0}$ | $-$
$\sigma_{8}$ | $-$
Table 1:  Model parameters and priors used in our cosmology analysis in flat
$\Lambda$CDM cosmology. $\mathcal{N}(\mu,\sigma)$ denotes the Gaussian
distribution with mean $\mu$ and standard deviation $\sigma$.
$\mathcal{U}(a,b)$ denotes the uniform distribution between the minimum value
$a$ and the maximum value $b$. The Gaussian priors on $\omega_{\mathrm{b}}$
and $n_{\mathrm{s}}$ are based on the BBN and Planck CMB constraints,
respectively (see text for details). The flat priors on other cosmological
parameters are set to be within the parameter ranges on which the emulator is
supported.

####  III.3.1 Cosmological parameters

Since we want to infer the cosmological parameters within the flat-geometry
$\Lambda$CDM framework, we sample all of the five parameters:

$\displaystyle\mathbf{p}_{\mathrm{cosmo}}=\left\\{\omega_{\mathrm{b}},\omega_{\mathrm{c}},\Omega_{\Lambda},\ln(10^{10}A_{\mathrm{s}}),n_{\mathrm{s}}\right\\},$
(16)

where $\omega_{\mathrm{b}}=\Omega_{\mathrm{b}}h^{2}$ and
$\omega_{\mathrm{c}}=\Omega_{\mathrm{c}}h^{2}$ are the physical energy density
parameters of baryon and cold dark matter, $\Omega_{\Lambda}$ is the energy
density parameter of the cosmological constant, and $A_{\mathrm{s}}$ and
$n_{\mathrm{s}}$ are the amplitude (at the pivot scale
$k_{\mathrm{pivot}}=0.05\,\mathrm{Mpc}^{-1}$) and the spectral tilt of the
power spectrum of primordial curvature perturbations. For
$\omega_{\mathrm{b}}$ and $n_{\mathrm{s}}$, we impose priors that are inferred
from other cosmological probes. More specifically, we adopt a Gaussian prior
on $\omega_{\mathrm{b}}$ from the primordial deuterium and helium abundance
data compared with the standard Big Bang nucleosynthesis (BBN) model [48, 66,
67]. On the other hand, we impose a Gaussian prior on $n_{\mathrm{s}}$ given
in Table 1 of Ref. [49] for the Planck 2018 “TT, TE, EE+lowE+lensing”. We
treat $\Omega_{\mathrm{m}}(=1-\Omega_{\Lambda})$, the total matter energy
density, $H_{0}=100h~{}\mathrm{km~{}s}^{-1}\mathrm{Mpc}^{-1}$, the Hubble
constant, and $\sigma_{8}$, the standard deviation of linear matter
perturbations at $z=0$ averaged within a sphere with comoving radius
$8\,h^{-1}\,\mathrm{Mpc}$, as the derived parameters. We fix the density
parameter of neutrinos to $\omega_{\nu}=0.00064$, corresponding to 0.06 eV for
the total mass of the three mass eigenstates, which is used when computing the
linear matter power spectrum used to set up the initial conditions of
cosmological simulations that are used for the Dark Emulator development [40].

####  III.3.2 Nuisance parameters

As we described in Sec. III.1, we employ five parameters to specify the HOD
model for each galaxy sample. In addition, we include two additional nuisance
parameters.

  * •

$c_{\mathrm{vel}}$ — The multiplicative coefficient on the velocity dispersion
of galaxies relative to the halo center. It regulates the uncertainty on the
strength of the FoG effect (see around Eq. 10).

  * •

$P_{\mathrm{shot}}$ — The residual shot noise contribution apart from the
simple Poisson shot noise. We add $P_{\mathrm{shot}}$ to the galaxy power
spectrum $P_{\mathrm{gg}}(k,\mu)$, and hence it is relevant only to the
monopole moment.

We employ a flat prior over the range
$[-10^{4},10^{4}]\,(h^{-1}\,\mathrm{Mpc})^{3}$ for the latter. The mean number
density of galaxies for each sample is a factor of a few times
$10^{-4}(h^{-1}\mathrm{Mpc})^{3}$, so the prior range is sufficiently wide.
Thus, we have seven nuisance parameters on the galaxy-halo connection:

$\displaystyle\mathbf{p}_{\mathrm{galaxy}}=\\{\log
M_{\mathrm{min}},\sigma_{\log M}^{2},\log
M_{1},\alpha_{\mathrm{sat}},\kappa,c_{\mathrm{vel}},P_{\mathrm{shot}}\\},$
(17)

for each of the 4 galaxy samples. Hence, the total number of parameters is
$5+7\times 4=33$ within the flat $\Lambda$CDM framework. As can be found from
Table 1, we employ a broad prior range for each of the galaxy-halo connection
parameters. For example, the range of $M_{\mathrm{min}}$, which is one of the
parameters that are sensitive to the linear bias of the galaxy sample,
corresponds to halos that have $b_{1}\simeq[1.2,10.2]$ at $z=0.5$ for the
Planck cosmology. Thus our approach can be considered conservative in
parameter inference.

###  III.4 Parameter inference

We employ the Bayesian inference to derive the parameter posterior
distribution:

$\displaystyle
p_{\mathrm{post}}(\mathbf{p}|\mathcal{D})\propto\mathcal{L}(\mathcal{D}|\mathbf{p})p_{\mathrm{prior}}(\mathbf{p}),$
(18)

where $p_{\mathrm{prior}}$ and $p_{\mathrm{post}}$ are the prior and posterior
distributions of model parameters, and $\mathcal{L}(\mathcal{D}|\mathbf{p})$
is the likelihood function of the observational data $\mathcal{D}$ given
parameters $\mathbf{p}$. Using the power spectrum data vector and the
covariance matrix, we compute the log-likelihood function:

$\displaystyle\ln\mathcal{L}(\mathcal{D}|\mathbf{p})=-\frac{1}{2}\sum_{\rm
samp}\sum_{\ell,\ell^{\prime}}\sum_{i,j}^{k_{\mathrm{max}}}\left[{P}^{\mathcal{D}}_{\ell}(k_{i})-{P}_{\ell}(k_{i};\mathbf{p})\right]$
$\displaystyle\,\,\,\times\mathrm{Cov}^{-1}\left[{P}_{\ell}(k_{i}),{P}_{\ell^{\prime}}(k_{j})\right]\left[{P}^{\mathcal{D}}_{\ell^{\prime}}(k_{j})-{P}_{\ell^{\prime}}(k_{j};\mathbf{p})\right],$
(19)

where we assume the Gaussian likelihood and omit the normalization factor.
${P}^{\mathcal{D}}_{\ell}(k_{i})$ denotes the data of the $\ell$-th multipole
moment of the power spectrum in the $i$-th wavenumber bin,
${P}_{\ell}(k_{i};\mathbf{p})$ is its theoretical model prediction, and
$\mathbf{p}$ is the model parameters. We include the power spectrum
information over $k_{\mathrm{min}}\leq k\leq k_{\mathrm{max}}$, and we employ
$k_{\mathrm{min}}=0.005\,h\,\mathrm{Mpc}^{-1}$ and
$k_{\mathrm{max}}=0.25\,h\,\mathrm{Mpc}^{-1}$ as our fiducial choices,
respectively. We will below give a validation of the choice of
$k_{\mathrm{max}}=0.25\,h\,\mathrm{Mpc}^{-1}$ and discuss how different
choices of $k_{\mathrm{min}}$ or $k_{\mathrm{max}}$ change the cosmological
results. The summation $\sum_{\rm samp}$ denotes the summation over galaxy
samples when the power spectrum information for different galaxy samples are
combined.

For the parameter sampling, we employ the Markov-chain Monte Carlo (MCMC)
sampling of the standard Metropolis algorithm [68], or the nested sampling
algorithm MultiNest [69] implemented in the public Python package PyMultiNest
[70]. In the MCMC sampling, we monitor the convergence for the cosmological
parameters by using a method in Ref. [71], which is an improved variant of the
Gelman-Rubin diagnostic [72, 73]. More specifically, we apply the rank-
normalization [71] to the MCMC chains and measure the Gelman-Rubin statistic
$\hat{R}$ of the chains split in half (so-called the split-$\hat{R}$ [74]),
after discarding 1000 points at the beginning of each chain as the burn-in
phase. We run the MCMCs until the criteria $\hat{R}<1.05$ for the cosmological
parameters are met. We use GetDist [75] package to draw triangle plots of
parameter posteriors.

We should note that, throughout this paper, we do not include the abundance of
galaxies (the mean number density) nor the BAO information after
reconstruction [e.g., see 76, for such a study] in the data vector in the
parameter inference.

Figure 1:  Results of validation tests of our cosmology analysis pipeline: we
apply the analysis pipeline to the mock signals for the 4 BOSS-like galaxy
samples using the covariance matrix of the actual BOSS spectra. We include the
monopole, quadrupole and hexadecapole moments of the redshift-space power
spectrum for each of the 4 galaxy samples over
$0.005\,h\,\mathrm{Mpc}^{-1}\leq k\leq k_{\mathrm{max}}$, and we show the
results for $k_{\mathrm{max}}=0.20,0.25$ or 0.30$\,h\,\mathrm{Mpc}^{-1}$,
respectively. The contours show the posterior distributions of
$\Omega_{\mathrm{m}},H_{0},$ and $\sigma_{8}$ including marginalization over
uncertainties in other parameters including the galaxy-halo connection
parameters. The left plot shows the results for the galaxy mocks that are
generated using a different recipe of the galaxy-halo connection from our
fiducial HOD model: the mock galaxy catalogs used in the Cosmology Challenges
in Ref. [37] (see text for details). Since we would like to blind the true
values of the cosmological parameters, we show the results in terms of the
parameter difference such as
$\Delta\Omega_{\mathrm{m}}=\Omega_{\mathrm{m}}-\Omega_{\mathrm{m,true}}$. The
right plot shows the results for the mock catalogs generated using the same
form of HOD model as in our HOD model. The dashed lines in the right panel are
the true values of the cosmological parameters used in the mock catalogs,
i.e., those for the Planck cosmology.

##  IV Results

We show the main results of our cosmology analysis in this section. Throughout
this paper, we mainly focus on the constraints on three cosmological
parameters $\Omega_{\mathrm{m}}$, $H_{0}$, and $\sigma_{8}$, which are well
constrained by the redshift-space galaxy power spectrum, in flat $\Lambda$CDM
model.

###  IV.1 Validation tests

Before showing the main results, we first present validation tests of our
emulator based method. To test the validity and usefulness of the emulator-
based method, we performed various cosmology challenges: we apply our
cosmology analysis pipeline to simulated mock signals of the redshift-space
galaxy power spectrum to address whether the pipeline can recover the
underlying true cosmological parameters used in the simulations. Please see
Ref. [37] for details of the procedures and aims. Y. K., who is one of the
authors of this paper and led the actual cosmology inference analysis of the
BOSS data, applied the pipeline to the mock signals for the BOSS-like galaxies
used in Ref. [37]. The mock galaxy catalogs were generated using a different
recipe for the galaxy-halo connection based on subhalos, so it is not entirely
clear whether our HOD method can recover the underlying cosmological
parameters. For instance, the spatial and velocity structures of satellite
galaxies in host halos are generally different from those in our fiducial halo
model. In this test, Y. K. was not informed of the cosmological parameters,
and the validation test was done effectively in a blind manner. He submitted
the results to T. N., who is a co-author of this paper and is the main
organizer and the maintainer of the challenge program, and then the ground
truth cosmological parameters were revealed with the mutual agreement not to
update the analysis anymore. The submitted results are recorded and presented
on the challenge webpage
(https://www2.yukawa.kyoto-u.ac.jp/~takahiro.nishimichi/data/PTchallenge/).

In Figure 1 we show the results of our validation tests. The left panel shows
the results using the simulated signals that are generated from the mock
catalogs of the above cosmology challenges of Ref. [37]. To mimic the
cosmology analysis of the BOSS power spectra, we use the mock catalogs at
$z=0.38$ and $0.61$ to simulate the redshift-space spectra for the 4
subsamples, the NGC/SGC in the low-$z$ and high-$z$ bins, and use the same
covariance matrix as that used in the following cosmology analysis of BOSS
spectra (see Sec. II). These mock catalogs are based on the realizations of
$N$-body simulations for the total volume of $566\,(h^{-1}\mathrm{Gpc})^{3}$
[37], which is about a hundred times that of the BOSS DR12 galaxy sample. The
large volume of the simulations allows us to sufficiently reduce the sample
variance errors in the simulated power spectra. Therefore, we can conduct a
fairly stringent test of the systematic error due to an imperfect modeling of
the power spectrum. The figure shows that our analysis method recovers the
true cosmological parameters to within the statistical errors of the BOSS
galaxy spectra. We also stress that our emulator-based method passes the
validation test even if including the power spectrum information up to
$k_{\mathrm{max}}=0.3\,h\,\mathrm{Mpc}^{-1}$, where the perturbation theory
breaks down. Here the constraints on $\Omega_{\rm m}$ and $H_{0}$ are mainly
from the BAO features and partly from the power spectrum shape via the AP
effect. On the other hand, the constraint on $\sigma_{8}$ is from the power
spectrum amplitude, after the degeneracies with galaxy bias uncertainty
(uncertainties in the galaxy-halo connection in our model) are lifted by
measurements of the RSD effect as we will discuss later (Sec. V.6). However,
including the information beyond $k\simeq 0.2\,h\,\mathrm{Mpc}^{-1}$ gives
little improvement in the cosmological parameters, due to the shot noise
domination and the degeneracies with the galaxy-halo connection parameters. On
the other hand, we find that the constraints on HOD parameters are improved by
including the higher-$k$ information.

As another sanity check, we also perform the validation tests using the mock
galaxy catalogs generated in Refs. [40, 17]. They are based on the simulations
for the Planck cosmology using the same HOD prescription as that in our
analysis. Similarly to the above test, we used the outputs of $N$-body
simulations at $z=0.251$ and 0.617, adopted the same HOD model as that in Ref.
[77] to populate galaxies into halos of each realization, and then simulated
the mock galaxy spectra for the 4 galaxy samples. We used the simulations of
total volume $128\,(h^{-1}\mathrm{Gpc})^{3}$ to generate the simulated data
vector [see 17, for details of the simulations], and performed the cosmology
analysis using the same covariance matrix of the BOSS spectra. The right panel
of Figure 1 shows that our analysis method recovers the cosmological
parameters to within the statistical errors, for all the $k_{\mathrm{max}}$
values. One might notice a slight bias in each cosmological parameter, even if
both the theoretical template and the mock catalogs employ the same form of
HOD model. We ascribe the parameter shift to the projection effect of the full
posterior distribution in a multi-dimensional parameter space [see 78, for the
similar discussion]. Here we note that some of the galaxy-halo connection
parameters are not necessarily recovered by the analysis as explicitly shown
in Figure 14 of Appendix A. The figure in the appendix also shows that
including the power spectrum information on the higher $k_{\mathrm{max}}$
gives smaller error bars for the galaxy-halo connection parameters, although
the central values are biased for some of the galaxy-halo parameters. On the
other hand, the credible intervals of the cosmological parameters are not much
improved by including the information from $k_{\rm max}=0.25$ to $0.30\,h{\rm
Mpc}^{-1}$, as can be found from Figure 1. Hence, from these results, we
conclude that our method can robustly recover the cosmological parameters to
within the statistical errors of the BOSS power spectra, after marginalization
over the galaxy-halo connection parameters. Here note that the cosmological
information is extracted from the redshift-space power spectrum of halos
predicted by the emulator in our method. In this paper, we employed
$k_{\mathrm{max}}=0.25\,h\,\mathrm{Mpc}^{-1}$ as our fiducial choice of the
maximum wavenumber.

We will also later show the validation test of our analysis method using the
mock catalogs including the assembly bias effect, which is one of the most
dangerous, physical systematic effects in the halo model approach.

Figure 2:  The posterior distributions of $\Omega_{\mathrm{m}},H_{0}$ and $\sigma_{8}$, obtained from the cosmology inference including the full shape of the monopole, quadrupole and hexadecapole moments of the BOSS DR12 galaxy power spectra up to $k_{\mathrm{max}}=0.25\,h\,\mathrm{Mpc}^{-1}$, for the flat $\Lambda$CDM model. For the theoretical templates we use the emulator-based halo model, and the posteriors include marginalization over uncertainties in other cosmological parameters and the nuisance parameters including the galaxy-halo connection parameters. For comparison, the red contours show the results from the Planck 2018 “TT, TE, EE+lowE” analysis.  Parameter | MAP | Median | 68% CI | Planck 68% CI
---|---|---|---|---
$\ln(10^{10}A_{\mathrm{s}})$ | 2.93 | 3.01 | $3.01_{-0.087}^{+0.089}$ | $3.045\pm 0.016$
$\Omega_{\mathrm{m}}$ | 0.300 | 0.302 | $0.301_{-0.011}^{+0.012}$ | $0.3166\pm 0.0084$
$H_{0}\,[\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}]$ | 68.3 | 68.2 | $68.2_{-1.4}^{+1.4}$ | $67.27\pm 0.60$
$\sigma_{8}$ | 0.754 | 0.786 | $0.786_{-0.037}^{+0.036}$ | $0.8120\pm 0.0073$
$S_{8}\equiv\sigma_{8}(\Omega_{\mathrm{m}}/0.3)^{0.5}$ | 0.754 | 0.787 | $0.784_{-0.042}^{+0.048}$ | $0.834\pm 0.016$
$f\sigma_{8}(z_{\rm eff}=0.38)$ | 0.474 | 0.471 | $0.467_{-0.028}^{+0.035}$ | $0.4771\pm 0.0066$
$f\sigma_{8}(z_{\rm eff}=0.61)$ | 0.434 | 0.434 | $0.430_{-0.026}^{+0.034}$ | $0.4696\pm 0.0053$
Table 2:  The results of the cosmological parameter inference of the BOSS
power spectra for $\Lambda$CDM model as in Figure 2. Note that we include the
BBN prior on $\omega_{\mathrm{b}}$ and the Planck CMB prior on the spectral
tilt $n_{\mathrm{s}}$ (see Table 1). For each parameter, we show the parameter
value at the maximum a posteriori (MAP), the median, and the mode with 68%
credible interval (CI) for the 1d posterior distribution of each parameter.
The parameters other than $\ln(10^{10}A_{\rm s})$ are derived parameters (see
Table 1). For comparison with constraints from other experiments, we also show
the constraints on $S_{8}\equiv\sigma_{8}(\Omega_{\rm m}/0.3)^{0.5}$ and
$f(z)\sigma_{8}(z)$, where $f$ is the linear growth rate $f\equiv\mathrm{d}\ln
D/\mathrm{d}\ln a$. For $f\sigma_{8}$, we show the results obtained from the
cosmology analysis using the subsample of low-$z$ NGC+SGC or high-$z$ NGC+SGC
sample at the effective redshift $z_{\rm eff}=0.38$ and 0.61, respectively
(see the right panel of Figure 5). For comparison, we also show the 68%
credible interval from the Planck 2018 “TT, TE, EE+lowE” analysis, taken from
Table 2 of Ref. [49].  Figure 3:  The comparison of monopole (blue),
quadrupole (red), and hexadecapole (green) moments between the data (symbols
with error bars) and the model predictions (solid lines) at MAP of the MCMC
chains in our cosmology analysis shown in Figure 2. The error bars are
computed from the diagonal elements of the covariance matrix, described in
Section II. Figure 4:  The median and 68% percentile interval of the HOD
functions for each of the BOSS galaxy samples (low-$z$/high-$z$ NGC/SGC
samples), obtained from the MCMC chains in Figure 2. The solid and dashed
lines are the medians of central+satellite and central-only HOD functions,
respectively.

###  IV.2 Results: $\Lambda$CDM cosmology

Figure 2 shows the main results of this paper, the cosmological parameters
obtained from the actual BOSS power spectra. The figure shows the projected
posterior distributions of $\Omega_{\mathrm{m}}$, $H_{0}$ and $\sigma_{8}$ for
the flat $\Lambda$CDM model, after marginalizing over the other parameters
such as the galaxy-halo connection parameters. These constraints include the
BAO and full shape information of the redshift-space power spectrum for the 4
samples of the low-$z$/high-$z$ NGC and SGC samples, up to
$k_{\mathrm{max}}=0.25\,h\,\mathrm{Mpc}^{-1}$ where the nonlinear effects such
as nonlinear bias, nonlinear clustering and nonlinear RSD are properly
included. Our method achieves precise measurements of the cosmological
parameters:

$\displaystyle\Omega_{\mathrm{m}}=0.301^{+0.012}_{-0.011}$ $\displaystyle
H_{0}=68.2\pm 1.4$ $\displaystyle\sigma_{8}=0.786^{+0.036}_{-0.037},$ (20)

where we report the mode and the 68% credible interval of the marginalized
posterior distribution for each parameter, respectively. Note that $H_{0}$ is
in units of $\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$.

Table 3 compares our result with those from the recent similar full-shape
analyses of the power spectrum multipoles using the PT-based theory models:
the EFTofLSS model [50] and the model based on Lagrangian perturbation theory
(LPT) [51]. They used the same power spectrum signals and window function of
BOSS DR12 provided by Ref. [52], i.e., the same dataset as that we use.  It is
interesting to note that our results are consistent with these two analyses to
within the statistical errors, even though our halo model-based method and the
PT-based ones are constructed based on totally different frameworks. The size
of the error bar in each parameter is comparable with those of the their
results. This might be counter-intuitive, because one might think that our
halo model based method is a more restrictive model than PT-based models, so
expect our method gives tighter constraints on the cosmological parameters.
Probably this is due to the wide priors of the galaxy-halo connection
parameters in our analysis, and we will later discuss a possible room of
improvements in the cosmological parameters within our method.

Parameter | This work |  EFTofLSS model [50] |  LPT model [51]
---|---|---|---
$\Omega_{\mathrm{m}}$ | $0.301_{-0.011}^{+0.012}$ | $0.312_{-0.012}^{+0.011}$ | $0.305\pm 0.01$
$H_{0}\,[\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}]$ | $68.2\pm 1.4$ | $68.5_{-1.3}^{+1.1}$ | $68.5\pm 1.1$
$\sigma_{8}$ | $0.786_{-0.037}^{+0.036}$ | $0.737_{-0.044}^{+0.040}$ | $0.738\pm 0.048$
Table 3:  Comparison of the parameter bounds from recent “full-shape” analyses
of the BOSS DR12 power spectrum ($68\%$ CI). For the analysis of the EFTofLSS
model, we quote the results using the window-convolved power spectrum (top row
of Table IV of Ref. [50]). All the results shown here are obtained from
exactly the same power spectrum data provided by Ref. [52].

We also note other works [27, 28, 79, 80, 81] that recently performed the
full-shape cosmology analysis of the BOSS power spectrum. However, these works
used the different dataset and/or analysis method from those in this paper, so
it is not easy to make an apple-to-apple comparison with our result. We here
give a note for completeness of our discussion, stressing that the full-shape
cosmology analysis is becoming possible to obtain cosmological constraints for
a given cosmological framework such as the flat $\Lambda$CDM model.

Our results can also be compared with the Planck 2018 CMB constraints: we
overplot the Planck 2018 cosmological constraints from the baseline likelihood
of “TT, TE, EE+lowE”; the MCMC chains of Planck 2018 cosmology analysis are
downloaded from the Planck Legacy Archive
(http://pla.esac.esa.int/pla/#cosmology), where the neutrino mass is fixed to
$0.06\,$eV as we did in our analysis. Our results are in good agreement with
the Planck results for all of the three cosmological parameters. In Table 2,
we also show the parameter value at the maximum a posteriori (MAP), the
median, and the mode with 68% credible interval (CI) for the 1d posterior
distribution of each parameter. For comparison with the weak lensing survey
results [e.g. 82, 57], we also give the results for $S_{8}$, the parameter on
which the weak lensing surveys can give most stringent constraint. In
addition, we show the results for $f(z)\sigma_{8}(z)$, which is the parameter
often used to characterize the constraint mainly from the RSD measurement on
linear scales. Note that, since we assume the flat $\Lambda$CDM cosmology, the
linear growth rate $f(z)$ is determined solely by $\Omega_{\mathrm{m}}$
through

$\displaystyle
f(z)=-\frac{3}{2}\Omega_{\mathrm{m}}(z)+\left[\int_{0}^{1}\frac{\mathrm{d}x}{\left\\{\Omega_{\mathrm{m}}(z)x^{-1}+(1-\Omega_{\mathrm{m}}(z))x^{2}\right\\}^{\frac{3}{2}}}\right]^{-1},$
(21)

where $\Omega_{\mathrm{m}}(z)=\Omega_{\mathrm{m}}(1+z)^{3}[H_{0}/H(z)]^{2}$ is
the time-dependent matter density parameter. It is in contrast to traditional
RSD analyses such as Ref. [26], where one performs model independent linear-
growth measurements, marginalizing over possible nonlinear corrections and
focusing only on the amplitude of the apparent anisotropies arising from the
linear RSD effect. Therefore, as we have already quoted earlier in this
subsection, we can determine fundamental cosmological parameters, breaking the
degeneracy between $\sigma_{8}$ and $f$ under the assumed cosmological model.
The table gives the constraints for $f(z)\sigma_{8}(z)$ that are obtained from
the MCMC analyses using either of the low-$z$ NGC+SGC sample or high-$z$
NGC+SGC sample at the effective redshift $z_{\rm eff}=0.38$ or $0.61$,
respectively (see below). Our constraint is from the combined information of
the BAO, the AP effect, the RSD effect and the amplitude and shape information
of the power spectrum under the flat $\Lambda$CDM framework.

Figure 3 shows that our model at MAP well reproduces all the measured
multipoles of redshift-space power spectra. The reduced chi-square value at
the MAP is $\chi^{2}/\mathrm{d.o.f.}\simeq 1.05$ for 267$(=300-33)$ degrees of
freedom (the $p$-value $p\simeq 0.28$), implying that the MAP model gives an
acceptable fit to the data. Taking a closer look at the figure, our model has
a slightly weaker BAO feature than data. This tendency reflects the fact that
our power spectrum emulator is based on the training data which have a larger
$k$-bin width ($\Delta k=0.02\,h\,\mathrm{Mpc}^{-1}$) than the data ($\Delta
k=0.01\,h\,\mathrm{Mpc}^{-1}$), to suppress the statistical scatters on the
training data (also see Appendix A of Ref. [38]). Since the BAO feature leads
to a tight constraint on $\Omega_{\mathrm{m}}$ through the AP effect, the
sharpening of the BAO feature in the emulator could improve the cosmological
constraints.

Figure 4 shows the mean HOD functions obtained from the MCMC chains. It can be
found that the galaxy population inferred from the BOSS DR12 galaxy power
spectra is almost confined to halos with masses, $M\gtrsim
10^{12}\,h^{-1}\,M_{\odot}$. This figure also shows that there is no
remarkable difference in the HOD among the 4 galaxy samples. Our results are
qualitatively consistent with the HODs estimated in the previous works [83,
84, 85]. We again stress that the HOD constraints are obtained from the
fitting of the emulator-based halo model to the redshift-space power spectrum,
without employing any strong priors on the HOD parameters nor using the
abundance information (the mean number density of galaxies). Hence, the
posterior distributions of HOD are purely from the redshift-space clustering
information, while the previous works took into account different clustering
information such as the galaxy-galaxy weak lensing and/or the projected
correlation function [e.g. 57, for such study].

##  V Discussion

In this section, we discuss the robustness of our cosmology results: we study
how different analysis methods and datasets change the inferred cosmological
parameters.

Figure 5:  The posterior distributions for cosmological parameters, obtained from the different galaxy subsamples. The left panel shows the results for 4 individual galaxy samples: low-$z$ NGC, low-$z$ SGC, high-$z$ NGC, and high-$z$ SGC. The right panel shows the results for each of low-$z$ or high-$z$ NGC+SGC samples. In the right panel, we overplot the gray contours to show the distribution for the full sample for comparison, which is the same as the blue contours in Figure 2.  | low-$z$ NGC | low-$z$ SGC
---|---|---
Parameter | MAP | Median | 68% CI | MAP | Median | 68% CI
$\ln(10^{10}A_{\mathrm{s}})$ | 3.05 | 3.10 | $3.12_{-0.14}^{+0.11}$ | 3.17 | 3.14 | $3.14_{-0.21}^{+0.20}$
$\Omega_{\mathrm{m}}$ | 0.311 | 0.318 | $0.317_{-0.015}^{+0.016}$ | 0.261 | 0.261 | $0.259_{-0.025}^{+0.028}$
$H_{0}\,[\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}]$ | 66.1 | 66.5 | $66.4_{-1.7}^{+1.8}$ | 71.1 | 73.0 | $72.4_{-3.6}^{+4.2}$
$\sigma_{8}$ | 0.782 | 0.817 | $0.816_{-0.051}^{+0.055}$ | 0.811 | 0.842 | $0.826_{-0.077}^{+0.102}$
$S_{8}\equiv\sigma_{8}(\Omega_{\mathrm{m}}/0.3)^{0.5}$ | 0.796 | 0.841 | $0.840_{-0.061}^{+0.063}$ | 0.756 | 0.783 | $0.763_{-0.084}^{+0.106}$
$f\sigma_{8}(z_{\rm eff}=0.38)$ | 0.457 | 0.481 | $0.481_{-0.033}^{+0.034}$ | 0.451 | 0.467 | $0.455_{-0.045}^{+0.061}$
| high-$z$ NGC | high-$z$ SGC
Parameter | MAP | Median | 68% CI | MAP | Median | 68% CI
$\ln(10^{10}A_{\mathrm{s}})$ | 2.65 | 2.76 | $2.78_{-0.193}^{+0.127}$ | 3.02 | 3.11 | $3.10_{-0.17}^{+0.18}$
$\Omega_{\mathrm{m}}$ | 0.298 | 0.296 | $0.297_{-0.020}^{+0.017}$ | 0.305 | 0.310 | $0.307_{-0.024}^{+0.028}$
$H_{0}\,[\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}]$ | 70.3 | 70.1 | $70.1_{-2.1}^{+2.2}$ | 67.6 | 67.6 | $67.7_{-3.1}^{+2.8}$
$\sigma_{8}$ | 0.685 | 0.717 | $0.697_{-0.044}^{+0.074}$ | 0.783 | 0.824 | $0.814_{-0.066}^{+0.079}$
$S_{8}\equiv\sigma_{8}(\Omega_{\mathrm{m}}/0.3)^{0.5}$ | 0.683 | 0.711 | $0.692_{-0.056}^{+0.084}$ | 0.789 | 0.838 | $0.824_{-0.079}^{+0.096}$
$f\sigma_{8}(z_{\rm eff}=0.61)$ | 0.393 | 0.410 | $0.399_{-0.027}^{+0.043}$ | 0.450 | 0.475 | $0.470_{-0.040}^{+0.047}$
Table 4:  Similar to Table 2, but this table shows the results obtained from
the cosmology analysis using each of the 4 galaxy samples.

###  V.1 Variations in the cosmological parameters for different galaxy
subsamples

In Figure 5 we study how the cosmological parameters are changed when using
different subsets of the data vector: the left panel shows the results for 4
individual galaxy samples, and the right panel shows the combined results for
each of the two redshift bins, where the two galactic hemispheres, NGC and
SGC, are combined. Shifts in each parameter display a similar trend to those
shown in Figure 2 of Ref. [27], even though we use a totally different
theoretical template, i.e., the halo model based method, compared to the
EFTofLSS in their paper. Hence, we think that the parameter shifts are likely
due to the sample variances in each subsample. For completeness of our
discussion, in Appendix B we show the 2d posterior distributions for the full
parameters for each of the 4 individual galaxy samples (see Figure 15). The
low-$z$ SGC sample displays a sizable difference in some parameters compared
to the other samples, but the difference is still within the statistical
errors. Hence, we cannot give any definite conclusion as to that sample could
have a potential observational systematics compared to the others.

###  V.2 The impact of hexadecapole moments

Figure 6:  The posterior distributions of cosmological parameters in our
fiducial cosmology analysis for the full galaxy sample, obtained if or not we
include the hexadecapole moments of the redshift-space power spectrum in the
parameter inference. The red contours are the same as those in Figure 2.

Figure 6 shows that the inclusion of the hexadecapole moment of the redshift-
space power spectrum yields only a subtle improvement in the cosmological
parameters, because the hexadecapole moments have lower signal-to-noise ratios
than the monopole and quadrupole moments as shown in Figure 3. However, we
note that the hexadecapole indeed improves some of the HOD parameters, which
is consistent with the finding in Ref. [86].

###  V.3 The cosmological information in the different range of $k$

Figure 7:  Comparison of the posterior distributions obtained when including
the redshift-space power spectrum information up to a different $k_{\rm max}$
in the analysis. We show the cases of $k_{\mathrm{max}}=0.2$ (blue), 0.25
(red), and 0.3 (green) $h\,\mathrm{Mpc}^{-1}$ and also show the results of the
Planck 2018 $\Lambda$CDM “TT, TE, EE+lowE” (gray). The cases of
$k_{\mathrm{max}}=0.25\,h\,\mathrm{Mpc}^{-1}$ and the Planck are identical to
those in Figure 2.

Figure 8:  The distribution of scatters in each cosmological parameter,
obtained from the cosmology analysis of noisy mock power spectra for the
low-$z$ and high-$z$ NGC samples, using the different $k_{\mathrm{max}}$ cuts;
the $x$-axis shows a shift in the mode value in the 1d posterior of each
parameter at $k_{\mathrm{max}}=0.25$ or 0.3 $h\,\mathrm{Mpc}^{-1}$, compared
to that at $k_{\mathrm{max}}=0.2\,h\,\mathrm{Mpc}^{-1}$. We used the 50
realizations of the noisy mock spectra, and the histogram in each panel
displays the distribution of parameter shifts. The vertical red and green
lines denote the shift found from the cosmology analyses of the real BOSS
data, shown in Figure 7. The dashed curves denote the Gaussian distributions
specified by the mean and variance of the parameter shifts among the 50
realizations.

One advantage of our approach, e.g., compared to the PT-based method such as
the EFTofLSS, is that ours enables to compare the model predictions with the
measurements up to higher $k_{\mathrm{max}}$. As described in our previous
paper [38], our emulator is designed to give an accurate prediction of the
redshift-space power spectrum up to $k=0.6\,h\,\mathrm{Mpc}^{-1}$. However the
power spectrum measurements at $k\gtrsim 0.2\,h\,\mathrm{Mpc}^{-1}$ are in the
shot noise dominated regime, and therefore it is not clear whether the
cosmological parameters are improved even if we include the data points at
higher wavenumbers. In Figure 7 we study how the cosmological parameters are
changed for different choices of $k_{\mathrm{max}}$; we consider
$k_{\mathrm{max}}$ $=0.2$, 0.25 or $0.3\,h\,\mathrm{Mpc}^{-1}$, respectively.
The figure shows that the size of the credible intervals is not largely
changed for the different $k_{\mathrm{max}}$ values, confirming the shot noise
domination in the power spectrum measurements at $k\gtrsim
0.2\,h\,\mathrm{Mpc}^{-1}$. We confirmed that the statistical errors of the
galaxy-halo connection parameters such as the residual shot noise and the HOD
parameters are indeed improved by including the information on the higher $k$.
A closer look reveals that the results for $k_{\mathrm{max}}=0.2$ and
$0.25\,h\,\mathrm{Mpc}^{-1}$ are consistent with each other. However, the
result for $k_{\mathrm{max}}=0.3\,h\,\mathrm{Mpc}^{-1}$ shows a sizable shift
in $\sigma_{8}$. For the validation tests using the noiseless mock signals in
Figure 1, we did not find this level of shift in the cosmological parameters.

As a further test of this shift, we use 50 realizations of noisy signals that
are generated by adding random noise realizations drawn from the BOSS
covariance matrix to the noiseless mock signals (the mock signals in the right
panel of Figure 1). Then we applied the same cosmology analysis pipeline to
each of the mock signals to estimate the cosmological parameters. Figure 8
shows the distributions of shifts in the cosmological parameters at
$k_{\mathrm{max}}=0.25$ or $0.30\,h\,\mathrm{Mpc}^{-1}$ with respect to those
at $k_{\mathrm{max}}=0.20\,h\,\mathrm{Mpc}^{-1}$, for the low-$z$ and high-$z$
NGC samples which give the dominant contributions to the cosmological
constraints of our full analysis. The figure shows that there is a reasonable
chance to have the parameter shifts for
$k_{\mathrm{max}}=0.25\,h\,\mathrm{Mpc}^{-1}$ seen from the actual BOSS data.
However, the shifts in some parameters for
$k_{\mathrm{max}}=0.3\,h\,\mathrm{Mpc}^{-1}$ are at the tail of the
distribution of the mock results, indicating a possible hint in the systematic
effects at $k\gtrsim 0.25\,h\,\mathrm{Mpc}^{-1}$, e.g., a limitation of the
halo model approach at such high $k$ scales or a residual systematic error in
the power spectrum data. Hence our fiducial choice of
$k_{\mathrm{max}}=0.25\,h\,\mathrm{Mpc}^{-1}$ seems reasonable against
possible systematic effects.

In Appendix C we also study possible effects of the fiber collision and the
minimum wavenumber $k_{\mathrm{min}}$ on the cosmological results. Here
$k_{\mathrm{min}}$ is the minimum wavenumber in that we include the power
spectrum information over $k_{\mathrm{min}}\leq k\leq k_{\mathrm{max}}$ in the
cosmology analysis. A brief summary is that these effects do not appear to
cause any major systematic effect in our cosmological results.

###  V.4 A further test of emulation accuracy

Now we turn to discussion on a possible uncertainty in the model predictions.
As discussed in Refs. [40] and [38], our emulator for the redshift-space halo
power spectrum is calibrated using a dataset of $N$-body simulations for the
101 flat $w$CDM models, where we used 15 realizations for the fiducial Planck
cosmology and one realization for each of 100 models (more exactly among these
we used the data for 80 models as training data). Here the 100 models are
sampled using the optimal maximum-distance sliced Latin hypercube design in
the 6-dimensional parameter space of $w$CDM cosmology [see 40, for details].
One might think that our emulator has a better accuracy around the Planck
cosmology, which is different from the MAP model or the model preferred by our
cosmology analysis of the BOSS power spectra. To test this possible
uncertainty, we ran a new set of $N$-body simulations for a model at the MAP
cosmology in Table 2: the model with
$\Omega_{\mathrm{m}}=0.300,H_{0}=68.3\,{\rm km~{}s}^{-1}{\rm Mpc}^{-1}$,
$\sigma_{8}=0.754$, and other cosmological parameters are set to the values at
MAP. More exactly, we ran each $N$-body simulation with box side length of
2.5$\,h^{-1}$Gpc and $3000^{3}$ particles, and use 5 realizations. The total
volume is about 78$\,(h^{-1}\mathrm{Gpc})^{3}$, much larger than the BOSS
volume ($\sim 5.7\,(h^{-1}\mathrm{Gpc})^{3}$), so we can sufficiently reduce
the sample variance effect in the simulated power spectra. Then we populate
galaxies into halos using the same recipe of the galaxy-halo connection used
in the Cosmology Challenge paper [37], which is different from our fiducial
HOD method. Using the same covariance matrix as we used in the actual
cosmology analysis of BOSS data, we perform the same cosmology analysis to the
simulated power spectra. This is very similar to the validation test of our
method using the simulated power spectrum used in the Cosmology Challenge
[37], but this new test gives a validation of our method at the MAP model.
Also note that the new set of $N$-body simulations employs the fixed neutrino
mass of $0.06\,\mathrm{eV}$ as in the simulations used for the emulator
development, while the simulations in the Cosmology Challenges assume zero
neutrino mass. Hence, this test also gives a confirmation on the subtle effect
of the non-zero neutrino mass, which mainly affects the transfer function of
matter density fluctuations. Figure 9 shows that our method recovers each
cosmological parameter accurately.

Figure 9:  The posterior distributions of the cosmological parameters using
the mock catalogs that are generated from the $N$-body simulations for a
cosmological model at MAP in Table 2 (see text for details). The horizontal
and vertical dashed lines show the input parameter values, i.e., the cosmology
at MAP. This is a similar test to Figure 1, and the mock catalogs are
generated using the same recipe of galaxy-halo connection as that in the left
panel of Figure 1.  Figure 10:  The posterior distributions of
$\Omega_{\mathrm{m}},H_{0}$, and $\sigma_{8}$, obtained when we apply our
analysis pipeline to the mock power spectrum signals that are generated from
the mock galaxy catalogs including the assembly bias effect (see text for the
details). The horizontal and vertical dashed lines show the input parameter
values. The mock galaxy catalogs have the same HOD shape as that of the mock
catalogs in the right panel of Figure 1, but were generated by populating mock
galaxies preferentially into halos that have lower concentrations, in each
halo mass bin.  Figure 11:  The red contours show the posterior distributions
of the galaxy-halo conection parameters for the low-$z$ NGC sample, obtained
for the same analysis using the assembly bias mock catalog as in Figure 10.
For comparison, the blue contours show the posterior distributions obtained
from the fiducial mock catalog, which are from the same analysis in the right
panel of Figure 1. The two mock catalogs are constructed using the same
galaxy-halo connection model, and the horizontal and vertical dashed lines
show the input parameter values. Figure 12:  Comparison of the multipole
moments between the measurements from the mock catalog and the model
predictions at MAP that are from the same analysis in Figure 11. The gray
shaded region is the $k$ range on which we use the data in the analysis, i.e.,
$k_{\mathrm{max}}=0.25\,h\,\mathrm{Mpc}^{-1}$.

###  V.5 Assembly bias

Another concern of the halo model approach is the impact of the “assembly
bias” effect; although the simple halo model assumes that the clustering
amplitudes of halos (and galaxies) are determined by halo masses, they might
depend on a secondary parameter, depending on the assembly history of
halos/galaxies [87, 88]. Even in the presence of the assembly bias, the
redshift-space distortion effect due to the peculiar velocities are unlikely
to be affected by the assembly bias, because the peculiar velocities arise
directly from the gravitational field [17]. To test a possible effect of the
assembly bias on cosmology analysis, we use the mock catalogs of galaxies
including the assembly bias effect in Ref. [17], where galaxies are populated
preferentially into halos that have lower concentrations as a proxy of the
assembly history [see 17, for details]. The mock catalogs including the
assembly bias effect are generated from the same $N$-body simulations as those
used in the right panel of Figure 1; therefore the total volume is about
$128\,(h^{-1}\,{\rm Gpc})^{3}$. The mock galaxies have about 30% higher
amplitudes in the real-space correlation function at large scales, compared to
that of the mock galaxies without the assembly bias effect, which otherwise
have the same HOD (see Figure 12). We generated the mock signals for each of
the BOSS-like galaxy samples, and applied the same pipeline of cosmology
analysis to the mock signals. Note that the assembly bias has not been
detected with high significance from the BOSS galaxies [e.g. 89].

Figure 10 shows the cosmological parameters obtained from the mock catalogs
including the assembly bias effect. It can be found that the assembly bias
does not cause a significant bias in the inferred cosmological parameters,
$\Omega_{\mathrm{m}},H_{0}$ and $\sigma_{8}$. This is consistent with the
Fisher forecast in the previous work [17], confirming that the BAO and RSD
information are not affected by the assembly bias effect, even after
marginalization over the galaxy-halo connection parameters. In other words,
the cosmology analysis using the redshift-space power spectrum does not rely
on the halo mass estimate. This result is contrasted with that in Ref. [77]:
they found that the assembly bias causes a significant bias in the
cosmological parameters, especially $\sigma_{8}$ and $\Omega_{\rm m}$, if a
hypothetical joint-probe cosmology analysis using the galaxy-galaxy weak
lensing and the projected correlation function is applied to the mock signals
including the assembly bias effect. In such a method, the galaxy-galaxy weak
lensing plays an important role to constrain the average mass of host halos,
which in turn helps determine the galaxy bias uncertainties to obtain the
clustering amplitude of matter at large scales. However, the assembly bias
disturbs the scaling relation of the large-scale bias amplitude with halo
mass, and in turn leads to biases in the inferred cosmological parameters. On
the other hand, our method using the redshift-space power spectrum does not
rely on the halo mass estimate. As can be found from Figure 11, some of the
galaxy-halo connection parameters are changed between the results for the
fiducial and assembly bias mocks, suggesting that the assembly bias effect are
absorbed by changes in the galaxy-halo connection parameters in our method.
For example, Figure 11 shows the model which has a slightly smaller
$\sigma_{\log M}$ than the input value is favored for the assembly bias mock,
which leads to a larger bias amplitude on large scales as explicitly shown in
Figure 12. The figure clearly shows that the model well reproduces the mock
measurements, especially the assembly bias effect that can be seen from the
higher amplitude of the monopole moment [also see Figure 16 in Ref. 17, for
similar discussion]. Hence we conclude that our cosmological results for the
BOSS galaxies are unlikely affected by the assembly bias, although the impact
of more general assembly bias effects on cosmology inference with the
redshift-space power spectrum further needs to be studied, e.g., using the
results in cosmological hydrodynamical simulations [e.g. 90].

###  V.6 Impact of uncertainty in the galaxy-halo connection

Finally we comment on the impact of the galaxy-halo connection parameters on
the cosmological parameters. We have so far employed broad priors of these
nuisance parameters. Other observables such as galaxy-galaxy weak lensing [85,
91] can be used to infer the HOD parameters. Throughout this paper, we have
employed a fairly broad prior range for each of 7 galaxy-halo connection
parameters. In this subsection, we study how the cosmological parameter
estimation can be improved if we have some knowledge on the HOD parameters.

Figure 13:  The posterior distributions of the cosmological parameters,
obtained from the cosmology analysis of the real BOSS data for either of the
low-$z$ or high-$z$ NGC sample, when fixing the 7 galaxy-halo connection
parameters to their values at MAP in Figure 15. The dashed-line contours and
1d posterior distribution are the same as in Figure 15.

As a working example, in Figure 13 we show the posterior distributions of the
cosmological parameters obtained from the BOSS data by fixing the galaxy-halo
connection parameters to their values at MAP in our fiducial analysis (the
results for Figure 2). Therefore, this can be regarded as the best-case
scenario where the galaxy-halo connection of the galaxy sample is perfectly
known. The figure shows that the cosmological parameters are significantly
improved. It is interesting to observe that the level of improvement of the
parameter constraint varies with the parameters. We can see that
$\Omega_{\mathrm{m}}$ and $H_{0}$, which are mostly determined by the
geometrical information through the BAO feature and also partly from the
spectral shape information, are not significantly improved as compared to
$\sigma_{8}$, which is an amplitude-related parameter. This is telling that
the overall amplitude of the cosmological fluctuations is the hardest to
interpret from observations due to the strong degeneracy with the galaxy bias
uncertainty or the uncertainty in galaxy-halo connection.

##  VI Conclusion

In this paper, we used an emulator of halo clustering statistics to estimate
the cosmological parameters from the full-shape analysis of the redshift-space
power spectra measured from the BOSS DR12 galaxy catalog over $0.2<z<0.75$.
Combining with the HOD model, the emulator allows us to compute the redshift-
space power spectra of galaxies for a given cosmological model within the flat
$\Lambda$CDM cosmology, in less than a CPU second. It enables the parameter
inference of the BOSS spectra in a multi-dimensional parameter space (33
parameters in this study). We showed that the emulator model well reproduces
the monopole, quadrupole and hexadecapole moments of the redshift-space power
spectra simultaneously for all of the 4 galaxy subsamples. Our method yields
stringent constraints on the cosmological parameters, $\Omega_{\mathrm{m}}$,
$H_{0}$ and $\sigma_{8}$, even after marginalization over uncertainties in the
nuisance parameters including the galaxy-halo connection parameters: more
precisely,  $\Omega_{\mathrm{m}}=0.301^{+0.012}_{-0.011}$, $H_{0}=68.2\pm
1.4~{}\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$, and
$\sigma_{8}=0.786^{+0.036}_{-0.037}$ , for the mode and the 68% credible
interval of the 1d posterior distribution (Figure 2 and Table 2). The
cosmological parameters we obtained are consistent with those from the
independent studies for the same BOSS spectra using PT-based models as the
theoretical template [50, 51], even though the theoretical templates are
totally different. This shows the robustness of the redshift-space power
spectrum method for estimating the cosmological parameters: the BAO, AP
effects and RSD information can be robustly extracted as long as the
uncertainties in the nonlinear effects and galaxy properties are marginalized
over. The statistical precision for each of the main parameters,
$\Omega_{\mathrm{m}}$, $H_{0}$ and $\sigma_{8}$ is also comparable with that
from the PT-based methods. One might think that the halo model could allow us
to use the redshift-space power spectra down to a larger $k$ and therefore
lead to a more stringent constraint on the parameters, compared to the PT-
based methods that treat all the galaxy bias parameters as nuisance
parameters. However, this is not the case. We think there are a few reasons
for this result. First, we employed quite broad priors for each of the HOD
parameters and did not include any information on the abundance of galaxies.
Hence our analysis might be considered a conservative approach. Second, the
power spectrum information at $k\gtrsim 0.2\,h\,\mathrm{Mpc}^{-1}$ is in the
shot noise dominated regime, and the cosmological parameters are not improved
even if including the power spectrum information on $k\gtrsim
0.2\,h\,\mathrm{Mpc}^{-1}$ (Figure 7). Our results are also in good agreement
with the Planck 2018 results [49] for $\Omega_{\mathrm{m}}$ and $H_{0}$, but
indicate a slight tension for $\sigma_{8}$ similarly to those reported by the
weak lensing analyses [82, 92, 93].

There is a promising route to improving our cosmological constraints. It is a
joint-probes cosmology: although we use the redshift-space power spectrum as
the data vector in this paper, there is another observable available in the
BOSS footprint. The promising one is the galaxy-galaxy weak lensing, which can
be obtained by cross-correlating the positions of BOSS galaxies with shapes of
background galaxies, where the background galaxies can be taken from an
imaging survey overlapping with some portion of the BOSS footprint. For this,
the Subaru HSC and some parts of the DES and KiDS surveys have an overlapping
region with the BOSS footprints, and the galaxy-galaxy weak lensing can be
measured from the joint analysis of BOSS and these imaging surveys. Since the
galaxy-galaxy weak lensing can measure the average mass distribution around
the BOSS galaxies, it helps observationally disentangle the galaxy bias
uncertainty or yields stringent constraints on the HOD parameters in the halo
model picture. In fact, Ref. [57] combined the galaxy-galaxy weak lensing,
measured from the HSC data of small area coverage 140 sq. degrees, with the
projected correlation function of BOSS galaxies in the cosmology analysis and
then obtained the accuracy of $\sigma(S_{8})\simeq 0.05$, compared to our
constraint of $\sigma(S_{8})\sim 0.04$ although $S_{8}$ is not a best
parameter to which the redshift-space power spectrum is sensitive. The galaxy-
galaxy weak lensing arises mainly from Fourier modes in the two-dimensional
space perpendicular to the line-of-sight direction, and it was shown that it
carries almost independent information from the redshift-space power spectrum
that arises from Fourier modes in the three-dimensional space [94]. Hence,
combining the redshift-space power spectrum with the galaxy-galaxy weak
lensing helps disentangle the degeneracy between the galaxy bias uncertainty
(the galaxy-halo connection) and the cosmological parameters, leading to an
improved estimation of the cosmological parameters (see Figure 13 for a
possible improvement in the best-case scenario). To do this, our emulator
based method easily enables to jointly combine the two observables in the same
halo model framework for parameter inference. This is definitely an
interesting direction, and will be our future work.

###### Acknowledgements.

We thank Mikhail Ivanov, Oliver Philcox, Shun Saito, Masato Shirasaki, Marko
Simonović, Naonori S. Sugiyama, Sunao Sugiyama, Ryuichi Takahashi, Digvijay
Wadekar, and Matias Zaldarriaga for useful discussions. After submission of
this paper to arXiv, we noticed the full-shape cosmology analysis of BOSS
power spectrum using the EFTofLSS method in Ref. [29]. The authors, Stephen
Chen and Martin White, kindly informed us of the updated measurements of BOSS
power spectrum done in Ref. [52]. We would like to thank them for useful
discussion. Y.K. also thanks to the Yukawa Institute for Theoretical Physics,
Kyoto University for the warm hospitality where this work was partly done.
This work was supported in part by World Premier International Research Center
Initiative, MEXT, Japan, and JSPS KAKENHI Grant No. JP15H03654, JP15H05887,
JP15H05893, JP15K21733, JP17H01131, JP17K14273, JP19H00677, JP20H01932,
JP20H04723, JP20H05850, JP20H05861, JP20H05855, and JP21H01081, by Japan
Science and Technology Agency (JST) CREST JPMHCR1414, by JST AIP Acceleration
Research Grant No. JP20317829, Japan, and by Basic Research Grant (Super AI)
of Institute for AI and Beyond of the University of Tokyo. Y.K. was also
supported by the Advanced Leading Graduate Course for Photon Science at the
University of Tokyo. The $N$-body simulations and subsequent halo-catalog
creation in the Dark Quest simulation suite used in this work were carried out
on Cray XC50 at Center for Computational Astrophysics, National Astronomical
Observatory of Japan.

##  Appendix A Posterior distributions of the galaxy-halo connection
parameters in the validation test

Figure 14:  The posterior distributions of the galaxy-halo connection
parameters, for the low-$z$ NGC sample as an example, obtained from the
combined analysis using the mock catalogs that are generated using the same
HOD model as in our theoretical template (the right panel of Figure 1). The
dashed lines in each plot denote the input value of each parameter that is
used when generating the mock galaxy catalogs.

Figure 14 show the posterior distributions of the galaxy-halo connection
parameters, obtained from the cosmology analysis of the mock data vector that
are generated from the mock catalogs of BOSS-like galaxies. The mock galaxy
catalog is the same as that used in the right panel of Figure 1. The figure
shows that some of the galaxy-halo connection parameters are not necessarily
well recovered, although the cosmological parameters are fairly well recovered
as shown in Figure 1. The size of the posterior contours for the galaxy-halo
connection parameters indeed shrinks with including the power spectrum
information on the higher $k$.

##  Appendix B Posterior distributions in full parameter space

Figure 15:  The posterior distributions in each 2d subspace of the full
parameters for each of the 4 galaxy samples, for the cosmology analysis in
Figure 2: the low-$z$ NGC, low-$z$ SGC, high-$z$ NGC and high-$z$ SGC samples,
respectively. The posterior distributions are from the joint parameter
inference of these parameters (33 parameters in total as given in Table 1): 5
cosmological parameters (while we impose tight Gaussian priors to
$\omega_{\mathrm{b}}$ and $n_{\mathrm{s}}$) and each sample is characterized
by 7 nuisance parameters (5 HOD parameters, the virial velocity parameter, and
the residual shot noise parameter).

For comprehensiveness of our discussion, in Figure 15 we show the posterior
distributions in each 2d subspace of the full parameters for the $\Lambda$CDM
model for each galaxy sample, based on our baseline analysis setup (Table 1).

##  Appendix C Tests of the fiber collision effect and the minimum wavenumber
$k_{\mathrm{min}}$

Next we investigate the impact of the fiber collision on the cosmological
parameter inference from the redshift-space galaxy power spectrum. The fiber
collision, which occurs due to the inability of the two adjacent optical
fibers to be closer than some finite separation angle, is a potential
systematic effect on the cosmological parameter inference. The likelihood of
the fiber collision depends on the number density of target galaxies on the
celestial sphere, and it leads to an anisotropic effect in the measured power
spectrum. Ref. [95] shows that the correction to the fiber collision effect on
the galaxy power spectrum depends on the true power spectrum itself, and
suggests the way for the correction. In this work, instead of implementing the
correction in the model prediction, we examine the extent to which the fiber
collisions affect the cosmology inference.

Figure 16:  The comparison of the cosmological parameter inferences between
the Patchy mock power spectra with and without the fiber collisions. We show
the cases of the low-z (blue) and high-z (red) NGCs and
$k_{\mathrm{max}}=0.3\,h\,\mathrm{Mpc}^{-1}$. The empty dashed-line contours
are the results for the Patchy mocks where we include the fiber collision
weights. The filled solid-line contours are those for the same mocks where we
assume all of the fiber collision weights are unity, i.e., no fiber
collisions. The gray horizontal and vertical dashed lines indicate the input
cosmological parameter values used in the simulations from which the Patchy
mocks are created.

Figure 16 shows a comparison of the parameter inference between the cases of
with/without the fiber collisions. To investigate the influence of the fiber
collisions on the cosmological parameter inference, we use the Patchy mocks
which have the fiber collision weights. The fiber collision weights in the
Patchy mocks are assigned following Ref. [54], which reflects the nearest
neighbor (NN) method. In this figure we see that there are only marginal
differences in the parameter posteriors between the case with the fiber
collisions corrected by the NN weights (empty, dashed line contours) and that
of the true power spectrum with no fiber collisions (filled, solid line
contours), for both the low-$z$ and high-$z$ NGCs, up to
$k_{\mathrm{max}}=0.3\,h\,\mathrm{Mpc}^{-1}$.

Figure 17:  The comparison of the cosmological parameter inferences between
the fiducial mock power spectrum with and without the fiber collision effect.
For mock signals with the fiber collisions, we add the fiber collision effect
estimated by Ref. [95] (see their Figure 3) to the power spectrum monopole and
quadrupole of the fiducial mock signals.

Figure 17 shows another test based on mock power spectra. We quote the shifts
of the power spectrum due to the fiber collisions 222We obtain this shifts
using WebPlotDigitizer (https://automeris.io/WebPlotDigitizer/) in the Nseries
mocks shown in the left panel of Figure 3 of Ref. [95], and add them to our
fiducial mock signals after spline interpolation to reduce the noise
contributions. This mock fiber collision effect causes almost no change to the
posterior distribution of the cosmological parameters, even in the case of
$k_{\mathrm{max}}=0.3\,h\,\mathrm{Mpc}^{-1}$. From these studies, we conclude
that the fiber collision effect has no significant impact on the cosmology
analysis. Therefore, we conclude that the systematic shift in the $\sigma_{8}$
posterior in our real data analysis when we increase $k_{\mathrm{max}}$ (shown
in Figure 7) is not due to the fiber collisions.

Figure 18:  The dependence of the cosmological parameter constraints on the
minimum wavenumber $k_{\mathrm{min}}$ used in the analysis. We tested the
cases of $k_{\mathrm{min}}=0.005$ (our fiducial setting, blue), 0.02 (red) and
$0.05\,h\,\mathrm{Mpc}^{-1}$ (green), while fixing
$k_{\mathrm{max}}=0.25\,h\,\mathrm{Mpc}^{-1}$.

We mention the influence of the minimum wavenumber $k_{\mathrm{min}}$ of the
power spectrum signals we use in the cosmological inference. Figure 18 shows
the parameter inference results for different values of
$k_{\mathrm{min}}=0.005,0.02$, and $0.05\,h\,\mathrm{Mpc}^{-1}$ while we keep
$k_{\mathrm{max}}=0.25\,h\,\mathrm{Mpc}^{-1}$. Here
$k_{\mathrm{min}}=0.005\,h\mathrm{Mpc}^{-1}$ is our default choice throughout
this paper. It shows that the choice of $k_{\mathrm{min}}$ has no significant
effect on the cosmological parameter inference.

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