# HSC Year 1 cosmology results with the minimal bias method: HSC$\times$BOSS
galaxy-galaxy weak lensing and BOSS galaxy clustering

Sunao Sugiyama<EMAIL_ADDRESS>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
Department of Physics, The University of Tokyo, Bunkyo, Tokyo 113-0031, Japan
Masahiro Takada<EMAIL_ADDRESS>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
Hironao Miyatake  Kobayashi-Maskawa Institute for the Origin of Particles and
the Universe (KMI), Nagoya University, Nagoya, 464-8602, Japan  Institute for
Advanced Research, Nagoya University, Nagoya 464-8601, Japan  Division of
Particle and Astrophysical Science, Graduate School of Science, Nagoya
University, Nagoya 464-8602, Japan  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  Jet
Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109,
USA  Takahiro Nishimichi  Center for Gravitational Physics, Yukawa Institute
for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan  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  Masato Shirasaki  National Astronomical
Observatory of Japan, Mitaka, Tokyo 181-8588, Japan  The Institute of
Statistical Mathematics, Tachikawa, Tokyo 190-8562, Japan  Yosuke Kobayashi
Department of Astronomy/Steward Observatory, University of Arizona, 933 North
Cherry Avenue, Tucson, AZ 85721-0065, USA  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  Surhud
More  The Inter-University Centre for Astronomy and Astrophysics, Post bag 4,
Ganeshkhind, Pune 411007, India  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
Ryuichi Takahashi  Faculty of Science and Technology, Hirosaki University, 3
Bunkyo-cho, Hirosaki, Aomori 036-8561, Japan  Ken Osato  Center for
Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto
University, Kyoto 606-8502, Japan  LPENS, Département de Physique, École
Normale Supérieure, Université PSL, CNRS, Sorbonne Université, Université de
Paris, 75005 Paris, France  Masamune Oguri  Research Center for the Early
Universe, The University of Tokyo, Bunkyo, Tokyo 113-0031, Japan  Department
of Physics, The University of Tokyo, Bunkyo, Tokyo 113-0031, Japan  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  Jean Coupon  Astronomy Department, University of
Geneva, Chemin d’Ecogia 16, CH-1290 Versoix, Switzerland  Chiaki Hikage  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  Bau-Ching Hsieh  Institute of Astronomy and
Astrophysics, Academia Sinica, Taipei 10617, Taiwan  Yotaka Komiyama  National
Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan  Alexie
Leauthaud  Department of Astronomy and Astrophysics, University of California,
1156 High Street, Santa Cruz, CA 95064, USA  Xiangchong Li  Department of
Physics, McWilliams Center for Cosmology, Carnegie Mellon University,
Pittsburgh, PA 15213, USA  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  Wentao Luo
Department of Astronomy, School of Physical Sciences, University of Science
and Technology of China, Hefei, Anhui 230026, China  Key Laboratory for
Research in Galaxies and Cosmology, School of Astronomy and Space Science,
University of Science and Technology of China, Hefei, Anhui 230026, China
Robert H. Lupton  Department of Astrophysical Sciences, Peyton Hall, Princeton
University, Princeton, NJ 08540, USA  Hitoshi Murayama  Department of Physics,
University of California, Berkeley, CA 94720, USA  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  Ernest Orlando Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  Atsushi J. Nishizawa  Institute for Advanced
Research, Nagoya University, Nagoya 464-8601, Japan  Youngsoo Park  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  Paul A. Price  Department of Astrophysical
Sciences, Peyton Hall, Princeton University, Princeton, NJ 08540, USA  Melanie
Simet  University of California Riverside, 900 University Ave, Riverside, CA
92521, USA  Jet Propulsion Laboratory, California Institute of Technology,
Pasadena, CA 91109, USA  Joshua S. Speagle  Banting & Dunlap Fellow
Department of Statistical Sciences, University of Toronto, Toronto, ON M5S
3G3, Canada  David A. Dunlap Department of Astronomy & Astrophysics,
University of Toronto, Toronto, ON M5S 3H4, Canada  Dunlap Institute for
Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada
Michael A. Strauss  Department of Astrophysical Sciences, Peyton Hall,
Princeton University, Princeton, NJ 08544, USA  Masayuki Tanaka  National
Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan

###### Abstract

We present cosmological parameter constraints from a blinded joint analysis of
galaxy-galaxy weak lensing, $\Delta\\!\Sigma(R)$, and projected correlation
function, $w_{\mathrm{p}}(R)$, measured from the first-year HSC (HSC-Y1) data
and SDSS spectroscopic galaxies over $0.15<z<0.7$. We use luminosity-limited
samples as lens samples for $\Delta\\!\Sigma$ and as large-scale structure
tracers for $w_{\mathrm{p}}$ in three redshift bins, and use the HSC-Y1 galaxy
catalog to define a secure sample of source galaxies at $z_{\mathrm{ph}}>0.75$
for the $\Delta\\!\Sigma$ measurements, selected based on their photometric
redshifts. For theoretical template, we use the “minimal bias” model for the
cosmological clustering observables for the flat $\Lambda$CDM cosmological
model. We compare the model predictions with the measurements in each redshift
bin on large scales, $R>12$ and $8~{}h^{-1}\mathrm{Mpc}$ for
$\Delta\\!\Sigma(R)$ and $w_{\mathrm{p}}(R)$, respectively, where the
perturbation theory-inspired model is valid. As part of our model, we account
for the effect of lensing magnification bias on the $\Delta\\!\Sigma$
measurements. When we employ weak priors on cosmological parameters, without
CMB information, we find $S_{8}=0.936^{+0.092}_{-0.086}$,
$\sigma_{8}=0.85^{+0.16}_{-0.11}$, and
$\Omega_{\mathrm{m}}=0.283^{+0.12}_{-0.035}$ (mode and 68% credible interval)
for the flat $\Lambda$CDM model. Although the central value of $S_{8}$ appears
to be larger than those inferred from other cosmological experiments, we find
that the difference is consistent with expected differences due to sample
variance, and our results are consistent with the other results to within the
statistical uncertainties. When combined with the Planck 2018 likelihood for
the primary CMB anisotropy information (TT,TE,EE+lowE), we find
$S_{8}=0.817^{+0.022}_{-0.021}$, $\sigma_{8}=0.892^{+0.051}_{-0.056}$,
$\Omega_{\mathrm{m}}=0.246^{+0.045}_{-0.035}$ and the equation of state
parameter of dark energy, $w_{\mathrm{de}}=-1.28^{+0.20}_{-0.19}$ for the flat
$w$CDM model, which is consistent with the flat $\Lambda$CDM model to within
the error bars.

††preprint: IPMU21-0078, YITP-21-126

##  I Introduction

Wide-area imaging and spectroscopic galaxy surveys provide us with powerful
tools for constraining the energy composition of the universe, the growth of
cosmic structure formation over time, and properties of the primordial density
perturbations (Weinberg et al., 2013). In particular, when combined with high-
precision measurements of the cosmic microwave background (Komatsu et al.,
2009; Planck Collaboration et al., 2016), galaxy surveys allow us to explore
the origin of the late-time cosmic acceleration, such as dark energy or a
possible breakdown of General Relativity on cosmological distance scales.
There are many existing, ongoing and upcoming galaxy surveys aimed at
advancing our understanding of these fundamental questions; e.g., the SDSS-
III/IV Baryon Acoustic Oscillation Spectroscopic Survey (BOSS/eBOSS) (Dawson
et al., 2016; Alam et al., 2021), the Subaru Hyper Suprime-Cam (HSC) survey
Aihara et al. (2018a), the Dark Energy Survey
(DES)111https://www.darkenergysurvey.org, the Kilo-Degree Survey
(KiDS)222http://kids.strw.leidenuniv.nl, the Subaru Prime Focus Spectrograph
(PFS) survey (Takada et al., 2014), the Dark Energy Spectroscopic Instrument
(DESI) survey333https://www.desi.lbl.gov, and then ultimately the Vera C.
Rubin Observatory Legacy Survey of Space and Time
(LSST)444https://www.lsst.org, Euclid555https://sci.esa.int/web/euclid, and
the Nancy Grace Roman Space Telescope666https://roman.gsfc.nasa.gov.

A major challenge in the use of galaxy surveys for precision cosmology lies in
uncertainties in the relationship between matter and galaxy distributions in
large-scale structure, i.e., uncertainties in the so-called galaxy bias
(Kaiser, 1984) (also see Desjacques et al., 2018, for a thorough review).
Since physical processes inherent in galaxy formation and evolution are still
very challenging to accurately model from first principles, we must
empirically model the galaxy bias and/or observationally constrain it. One
promising observational approach is a joint probes cosmological analysis
combining galaxy-galaxy weak lensing and galaxy clustering (Seljak et al.,
2005; Cacciato et al., 2009; Mandelbaum et al., 2013; Cacciato et al., 2012;
Hikage et al., 2013; More et al., 2015; Krause et al., 2017; Abbott et al.,
2018; van Uitert et al., 2018; Wibking et al., 2020; Nicola et al., 2020;
Miyatake et al., 2021). The two-point correlation function, $\xi_{\rm gg}$, is
the standard tool to characterize the large-scale structure through galaxy
clustering (Peacock et al., 2001; Tegmark et al., 2004). For a cold dark
matter-dominated universe with adiabatic, Gaussian initial conditions, the
galaxy correlation function is related to the two-point correlation function
of the underlying matter distribution, $\xi_{\rm mm}$, on large scales via a
linear bias parameter as $\xi_{\rm gg}(r)\simeq b_{1}^{2}\xi_{\rm mm}(r)$
(Kaiser, 1984), where $b_{1}$ is a scale-independent coefficient, the value of
which depends on galaxy properties (Dalal et al., 2008). Cross-correlating the
positions of galaxies with the shapes of background galaxies as a function of
their separations on the sky provides a measurement of the average matter
distribution around the foreground (lensing) galaxies – the so-called galaxy-
galaxy weak lensing (Mandelbaum et al., 2005, 2006). The galaxy-galaxy weak
lensing arises from the galaxy-matter cross-correlation, $\xi_{\rm gm}$, which
is given, at large scales, by $\xi_{\rm gm}(r)\simeq b_{1}\xi_{\rm mm}(r)$.
Hence, combining $\xi_{\rm gg}$ and $\xi_{\rm gm}$ allows one to
observationally constrain the galaxy bias for the foreground galaxy sample, at
least on large scales.

On the theory side, providing sufficiently accurate theoretical templates for
accurately extracting cosmological information from the clustering observables
remains difficult. There are two competing goals for the analysis: –
“robustness” and “precision” (also see Refs. Nishimichi et al., 2020; Miyatake
et al., 2021, for a study based on similar motivation). Achieving robust
results requires us to minimize any possible bias or shift in the estimated
value(s) of cosmological parameter(s) from the true value(s). On the other
hand, achieving precise results involves obtaining as small of a credible
interval (error bars) in cosmological parameters as is possible from the
observables. Obviously it is not straightforward to achieve these two goals
simultaneously. For example, since the galaxy clustering observables have a
higher signal-to-noise ratio at smaller spatial scales, which are affected by
nonlinear structure formation and galaxy physics, increasing the precision
(reducing the error bars) in cosmological parameters requires use of the
clustering observables down to small scales in the nonlinear regime. If the
theoretical model is not sufficiently accurate on these small scales, it can
easily lead to a large bias in the estimated cosmological parameters. The
worst case scenario is that one could measure cosmological parameters that
differ at high significance from the true ones.

In this paper we show the results of cosmological parameter estimation from
joint analysis of the galaxy-galaxy weak lensing, $\Delta\\!\Sigma(R)$, and
the projected correlation function of galaxies, $w_{\rm p}(R)$, measured from
the Subaru HSC Year 1 data (hereafter HSC-Y1 and see Refs. Aihara et al.,
2018a, b, for details) and the spectroscopic LOWZ and CMASS galaxy samples
from the Sloan Digital Sky Survey (SDSS Dawson et al., 2013). For the
theoretical template, we use the “minimal” bias model motivated by the
perturbation theory of structure formation (Bernardeau et al., 2002); we model
the galaxy-matter cross-correlation and the galaxy auto-correlation using a
linear bias parameter and the nonlinear matter power spectrum. Sugiyama et al.
(2020) evaluated the performance of this method by comparing the model
predictions with simulated $\Delta\\!\Sigma(R)$ and $w_{\rm p}(R)$ signals
mimicking the HSC-Y1 and SDSS measurements. They showed that the model can
recover the underlying cosmological parameters to within the statistical
errors as long as the parameter inference is restricted to relatively large
scales, $R\gtrsim 10\,h^{-1}{\rm Mpc}$, where the cross-correlation
coefficient, defined as $r_{\rm cc}(r)\equiv\xi_{\rm gm}(r)/[\xi_{\rm
gg}(r)\xi_{\rm mm}(r)]^{1/2}$, is close to unity for a $\Lambda$CDM-like
cosmology.

In our cosmological inference we perform a blind analysis to avoid
confirmation biases affecting our results. After unblinding, we compare our
results with the results from other cosmological experiments such as Planck
and other weak lensing surveys. The results of this paper can be compared with
the results in our companion paper, Miyatake et al. (2021), which infers
cosmological parameters by applying the halo model to exactly the same
observables while including smaller-scale information than in this paper.

This paper is organized as follows. We briefly review the HSC-Y1 and SDSS data
and catalogs used in this paper in Section II, and then describe the
measurements of $\Delta\\!\Sigma$ and $w_{\rm p}$ in Section III. Here we
refer to Miyatake et al. (2021) for the details. In Section IV we describe our
cosmological analysis method: the theoretical templates, the likelihood
analysis, and the analysis setup. In Section V we describe the blinding scheme
for our cosmological analysis. In Section VI we show the resulting
cosmological parameter constraints. Finally, we conclude in Section VII.

Throughout this paper we quote 68% credible intervals for parameter
uncertainties unless otherwise stated.

##  II The SDSS and HSC-Y1 data: large-scale structure tracers and source
galaxies

We use the data from the first-year Subaru Hyper Suprime-Cam survey (hereafter
HSC or HSC-Y1; Aihara et al., 2018a, b) and the SDSS-III BOSS DR11
spectroscopic sample of galaxies 777https://www.sdss.org/dr11/. (Alam et al.,
2015; Dawson et al., 2013). Hence we refer readers to Miyatake et al. (2021)
for details, and here we briefly review the most essential aspects for this
paper.

This paper focuses on cosmological parameter inference from joint measurements
of galaxy-galaxy weak lensing ($\Delta\\!\Sigma$) and the projected galaxy
auto-correlation function ($w_{\rm p}$). As tracers of large scale structure,
we use luminosity-limited samples selected from the SDSS spectroscopic galaxy
sample, after performing k-corrections. We consider three galaxy samples
covering three distinct redshift ranges: “LOWZ” galaxies in the redshift range
$z=[0.15,0.35]$ and two subsamples of “CMASS” galaxies, hereafter called
“CMASS1” and “CMASS2”, respectively. These are obtained by subdividing CMASS
galaxies into two redshift bins, $z=[0.43,0.55]$ and $[0.55,0.70]$,
respectively. More precisely, we select SDSS galaxies with absolute $i$-band
magnitudes $M_{i}\leq-21.5$, $-21.9$ and $-22.2$ for the LOWZ, CMASS1 and
CMASS2 samples, respectively, yielding comoving number densities $\bar{n}_{\rm
g}/[10^{-4}\,(h^{-1}{\rm Mpc})^{-3}]\simeq 1.8$, $0.74$ and $0.45$ for the
$\Lambda$CDM model that is consistent with the Planck 2015 “TT,TE,EE+lowP”
constraints. These are lower than those of the full LOWZ and CMASS samples by
a factor of a few. The luminosity-selected samples will allow for a higher-
fidelity cosmology analysis because they minimize possible redshift evolution
of galaxy properties within the redshift bin (e.g. Miyatake et al., 2015).

We use the HSC-Y1 galaxy catalog to define a secure sample of source galaxies
behind the lens galaxies in each SDSS galaxy sample, for galaxy-galaxy weak
lensing measurements. In this paper we use the HSC-Y1 shape catalog of
galaxies (Mandelbaum et al., 2018a), in combination with photo-$z$ information
(Tanaka et al., 2018). While multiple photo-$z$ catalogs based on different
methods are available, we use the catalog based on the MLZ method as our
fiducial catalog. The depth of HSC-Y1 data permits us to securely select
background galaxies behind the SDSS galaxies. We define a sample of background
galaxies by imposing the following cut for each HSC galaxy:

$\displaystyle\int^{7}_{z_{\rm l,max}+0.05}\mathrm{d}z_{\rm s}~{}P_{i}(z_{\rm
s})\geq 0.99,$ (1)

where $P_{i}(z_{\rm s})$ is the posterior distribution of photo-$z$ for the
$i$-th HSC galaxy. Here $z_{\rm s}=0.75$ is chosen for the lower bound of the
integration so that the HSC source galaxies are at redshifts greater than the
highest redshift of SDSS galaxies in the CMASS2 sample by more than $\Delta
z=0.05$, with probability greater than 0.99. For the upper bound we set
$z_{\rm s}=7$, the maximum redshift adopted in the HSC photo-$z$ catalog. We
use 4,308,983 HSC galaxies over about 140 deg2 in total, corresponding to
$\bar{n}_{\rm s}\simeq 8.74\,{\rm arcmin}^{-2}$ for the net number density or
$\bar{n}_{\rm s}\simeq 7.95\,{\rm arcmin}^{-2}$ for the weighted number
density (see Ref. Chang et al., 2013, for the definition). The unweighted mean
redshift of the sample is $\langle z_{\rm s}\rangle\simeq 1.34$.

In this paper we adopt a single population of source galaxies for galaxy-
galaxy weak lensing measurements following the method in Oguri and Takada
(2011); in other words, we do not select source galaxies separately for each
lens sample. Comparing the relative amplitudes of $\Delta\\!\Sigma$ for the
spectroscopic lens samples at different redshifts, while using the same source
sample, allows for a calibration of the average redshift of the HSC source
galaxies, i.e. a self-calibration of the photo-$z$ errors. In addition, it
allows for a self-calibration of the multiplicative shear biases that may
remain in the HSC shape catalog, because they cause redshift-independent
shifts in the $\Delta\\!\Sigma$ amplitudes for all the lens samples. Moreover,
the use of a single source population also allows for relatively
straightforward treatment of the magnification bias effect on the galaxy-
galaxy weak lensing, as we will discuss later.

##  III Measurements

Figure 1: The galaxy-galaxy lensing signal, $\Delta\\!\Sigma(R)$, measured by
combining the SDSS spectroscopic galaxies and the HSC photometric galaxies for
lens and source galaxy samples, respectively. For illustrative purpose, we
show $R\times\Delta\\!\Sigma(R)$ so that the dynamic range of the $y$-axis is
narrower. In this paper we consider three luminosity-limited lens samples,
LOWZ, CMASS1 and CMASS2, in the redshift ranges $z=[0.15,0.35]$, $[0.43,0.55]$
and $[0.55,0.70]$, respectively (see text for details). The un-shaded region
shows the range of separations used for cosmological parameter estimation in
this paper: $R=[12,80]h^{-1}\mathrm{Mpc}$. The label in the lower left corner
of each panel gives the cumulative signal-to-noise (S/N) ratio over the
fitting range for each galaxy sample. The total S/N given in the upper right
corner of the right panel accounts for the cross-covariances between the
$\Delta\\!\Sigma$ signals for the different lens samples. The red line in each
panel indicates the mode of the posterior distributions of the model
predictions in each separation bin, which are obtained from Bayesian inference
applied to the $\Delta\\!\Sigma$ and $w_{\rm p}$ data vectors for the three
galaxy samples assuming a flat $\Lambda$CDM model. The dark and light red-
colored regions are the 68% and 95% credible intervals of the posterior
distribution in each bin. The posterior distributions include marginalization
over other parameters. The blue line displays the model prediction at maximum
a posteriori (MAP) in the Bayesian inference.  Figure 2: Similar to the
previous figure, but this figure shows the projected auto-correlation
functions, $R\times w_{\rm p}(R)$, measured using the same LOWZ, CMASS1 and
CMASS2 lens galaxy samples as in the previous figure. The un-shaded region
denotes the range of separations used for cosmological parameter estimation:
$R=[8,80]h^{-1}\mathrm{Mpc}$.

The details of the measurement methods for the clustering observables,
$\Delta\\!\Sigma(R)$ and $w_{\rm p}(R)$, can be found in Miyatake et al.
(2021).

For the $\Delta\\!\Sigma(R)$ measurements, we measure the average shapes of
background HSC galaxies around each SDSS lens galaxy using the overlapping
regions between the HSC-Y1 and SDSS survey footprints, which cover about 140
deg2. The galaxy-galaxy weak lensing measures the average excess surface mass
density profile around the SDSS galaxies in each redshift bin. The measurement
includes a correction for the lensing efficiency, $\Sigma_{\rm cr}(z_{\rm
l},z_{\rm s})$, and an estimate of the projected separation $R$ from the
angular separation for each lens-source pair, which requires us to assume a
reference cosmology that generally differs from the underlying true cosmology.
For a flat $w\Lambda$CDM cosmology, the relevant cosmological parameters for
the conversion are $\Omega_{\rm m}$ and the dark energy equation of state
parameter $w_{\rm de}$. We use the method in Ref. More (2013) to account for
the $\Omega_{\rm m}$ and $w_{\rm de}$ dependences of the $\Delta\\!\Sigma$
measurement. Note that, once the $\Sigma_{\rm cr}$ factor is included, the
observable $\Delta\\!\Sigma(R)$ depends only on the clustering properties of
the SDSS lens galaxies, and does not depend on the redshifts of HSC source
galaxies.

For the $w_{\rm p}(R)$ measurements, we use the Landy & Szalay estimator
(Landy and Szalay, 1993) to estimate the three-dimensional clustering
correlation function of $\xi_{\rm gg}(R,\Pi)$ for each SDSS galaxy sample
using the entire SDSS footprint covering about 8,000 deg2, and then project
$\xi_{\rm gg}(R,\Pi)$ over $\Pi=[-\pi_{\rm max},\pi_{\rm max}]$ to obtain the
projected auto-correlation function $w_{\rm p}(R)$. We adopt $\pi_{\rm
max}=100\,h^{-1}{\rm Mpc}$ as our default choice. The projected correlation
function is not very sensitive to the redshift-space distortion (RSD) due to
peculiar motions of galaxies, but the RSD effect is not completely negligible
either (van den Bosch et al., 2013; Miyatake et al., 2020). We use the method
in Ref. van den Bosch et al. (2013) to include the RSD effect on $w_{\rm
p}(R)$ and its cosmological dependence (see also Sugiyama et al., 2020).

Figs. 1 and 2 show the $\Delta\\!\Sigma(R)$ and $w_{\rm p}(R)$ signals
measured for the LOWZ, CMASS1 and CMASS2 samples. In this paper we use the
signals over the range of $R$ indicated by the un-shaded region for our
cosmology analysis: we use the measurements over $12<R/[h^{-1}{\rm Mpc}]<80$
(8 bins) for $\Delta\\!\Sigma(R)$ and $8<R/[h^{-1}{\rm Mpc}]<80$ (14 bins) for
$w_{\rm p}(R)$, respectively. Note that we do not include information from the
baryon acoustic oscillations in $w_{\rm p}(R)$ at $R\sim 100\,h^{-1}{\rm Mpc}$
for the cosmological parameter inference. The red-colored regions in each
panel indicate the 68% and 95% intervals of the posterior distributions of the
model predictions in each separation bin, which we obtain from the
cosmological parameter inference using the measurements of $\Delta\\!\Sigma$
and $w_{\rm p}$ for all three samples assuming a flat $\Lambda$CDM model. The
blue line indicates the model prediction for the maximum a posteriori (MAP) in
the parameter estimation. It is clear that the model prediction can fairly
well reproduce all of the measurements simultaneously. However, one might
notice a sizable discrepancy in $\Delta\\!\Sigma$ between the MAP model
prediction and the measurement, for the LOWZ sample. The measured signals at
$R\gtrsim 30\,h^{-1}{\rm Mpc}$ are systematically lower than the model
prediction by more than $1\sigma$ errors, although the data points are highly
correlated with each other at these scales. We will below discuss in detail
how the cosmological constraints are changed if the LOWZ sample is removed
from the data vector in the parameter inference.

To carry out the cosmological parameter inference, we must model the
covariance matrices that describe statistical uncertainties in the
$\Delta\\!\Sigma(R)$ and $w_{\rm p}(R)$ measurements. We use the jackknife
method to estimate the covariance for $w_{\rm p}$. On the other hand, we use
mock catalogs of SDSS lens galaxies and HSC source galaxies to model the
covariance for $\Delta\\!\Sigma$, as described in Appendix B of Miyatake et
al. (2021). The covariance matrix for the $\Delta\\!\Sigma$ data vector
includes cross-correlations between the $\Delta\\!\Sigma$ signals for
different lens sample, which arise from the shape noise from the same source
galaxies and the cosmic shear on the same source galaxies due to the shared
foreground large-scale structure. In this and companion papers Miyatake et al.
(2021), we also include a contribution of the magnification bias to the
covariance matrix of $\Delta\\!\Sigma$, as derived in Appendix A. We neglect
the cross-covariance between $\Delta\\!\Sigma$ and $w_{\rm p}$, because the
overlapping area ($\sim$140 deg2) between the HSC-Y1 and SDSS survey
footprints is very small compared to the SDSS survey area ($\sim 8,000$ deg2).

In the legend of Figs. 1 and 2, we give the total signal-to-noise ratio (S/N)
that is obtained by integrating the S/N at each separation bin over the
fitting range, $R=[12,80]$ or $[8,80]\,h^{-1}{\rm Mpc}$ for $\Delta\\!\Sigma$
and $w_{\rm p}$, respectively, while accounting for the cross covariances
between the different $R$ bins. For the total S/N for $\Delta\\!\Sigma$, we
further take into account the cross covariances between the different
$\Delta\\!\Sigma(R)$ signals for the different lens samples. The S/N values
for $w_{\rm p}$ are higher than those for $\Delta\\!\Sigma$ for each SDSS
sample, meaning that our cosmological constraints are dominated by $w_{\rm
p}(R)$. However, combining it with galaxy-galaxy weak lensing,
$\Delta\\!\Sigma(R)$, is critical to constrain cosmological parameters,
because the combination helps break degeneracies between the galaxy bias and
cosmological parameters that determine the underlying matter clustering. The
total S/N of the combined (all three samples) $\Delta\\!\Sigma$ data vector is
10.5, while the total S/N of the combined $w_{\rm p}$ data vector is 47.5.

##  IV Analysis method

In this section, we describe our cosmological parameter inference methods
using the measured $\Delta\\!\Sigma(R)$ and $w_{\rm p}(R)$ signals described
in the preceding section.

###  IV.1 Theoretical model

In this paper we adopt the “minimal bias” model to interpret the measured
$\Delta\\!\Sigma(R)$ and $w_{\rm p}(R)$ signals (see Sugiyama et al., 2020,
for details). This model is the simplest one because it models
$\Delta\\!\Sigma(R)$ and $w_{\rm p}(R)$ using a linear bias parameter and the
nonlinear matter correlation function. Sugiyama et al. (2020) demonstrated and
validated this method by comparing theoretical templates with mock
$\Delta\\!\Sigma$ and $w_{\rm p}$ signals that mimic those of the SDSS
galaxies used in this paper, showing that this method can recover the
underlying cosmological parameters to within the statistical errors for the
$\Lambda$CDM cosmological model that is consistent with the Planck data(Planck
Collaboration et al., 2016), as long as the analysis is restricted to
clustering information on large scales above $R_{\rm cut}=12$ and
$8\,h^{-1}{\rm Mpc}$ for $\Delta\\!\Sigma$ and $w_{\rm p}$, respectively. To
be more quantitative, Sugiyama et al. (2020) showed that a shift in the
central value in $S_{8}$ from its true value is found to be within $\sim
0.5\sigma$ ($\sigma$ is the marginalized 68% statistical error) for all the
mock signals, one of which is implemented with extremely large assembly bias
effect and has a factor of 1.5 larger amplitude in $w_{\rm p}$ compared to
that of the fiducial mock. The results of this paper using the minimal bias
model can be compared with those of our companion paper, Miyatake et al.
(2021), which adopts the halo model approach to model the smaller-scale
clustering signals of the SDSS galaxies down to $(2,3)~{}h^{-1}{\rm Mpc}$ for
$\Delta\\!\Sigma$ and $w_{\rm p}$, respectively.

Using the minimal bias model we model $\Delta\\!\Sigma(R;z_{\rm l})$ for each
of the LOWZ, CMASS1 and LOWZ2 samples as

$\displaystyle\Delta\\!\Sigma(R;z_{\rm l})=b_{1}(z_{\rm l})\bar{\rho}_{\rm
m0}\int_{0}^{\infty}\\!\frac{k\mathrm{d}k}{2\pi}~{}P^{\rm NL}_{\rm
mm}(k;z_{\rm l})J_{2}(kR),$ (2)

where $J_{2}(x)$ is the 2nd-order Bessel function, $\bar{\rho}_{\rm m0}$ is
the mean matter density today, $b_{1}(z_{\rm l})$ is the linear galaxy bias
parameter for each SDSS galaxy sample at the redshift $z_{\rm l}$, and $P_{\rm
mm}^{\rm NL}$ is the nonlinear matter power spectrum at $z_{\rm l}$. In this
paper, we use the halofit fitting formula (Takahashi et al., 2012; Smith et
al., 2003) to model $P^{\rm NL}_{\rm mm}$ for an assumed cosmological model.

Throughout this paper we model the clustering observables at a representative
redshift of each SDSS lens sample, defined by the mean redshift of the lens
galaxies within each redshift bin. In other words, we do not include redshift
evolution of the clustering observables within the redshift range. We employ
$z_{\rm l}=0.26,0.51$ and $0.63$ for the representative redshifts of the LOWZ,
CMASS1 and CMASS2 samples, respectively. We checked that the difference
between the signal evaluated at the representative redshift and the signal
averaged over the redshift bin is below 4% of the square root of the diagonal
element of the covariance matrix in each $R$ bin, and hence this treatment
should not cause any significant bias in cosmological parameters. In the
following we omit the argument $z_{\rm l}$, e.g., in $\Delta\\!\Sigma(R;z_{\rm
l})$, for notational simplicity.

The large-scale structure between us and the lens galaxies distorts the shapes
of the background source galaxies and modulates their number density. The same
large-scale structure also causes number density fluctuations in the lens
sample due to lensing magnification; these correlate with the source galaxy
shape distortions (Unruh et al., 2020; von Wietersheim-Kramsta et al., 2021).
This effect adds a contamination term to the standard galaxy-galaxy weak
lensing, expressed as

$\displaystyle\Delta\\!\Sigma^{\rm
obs}(R)=\Delta\\!\Sigma(R)+\Delta\\!\Sigma^{\rm mag}(R).$ (3)

The first term on the r.h.s. is the standard galaxy-galaxy weak lensing
contribution (Eq. 2) and the second term is the contamination due to
magnification bias, which is expressed in terms of the nonlinear matter power
spectrum as

$\displaystyle\Delta\\!\Sigma^{\rm mag}(R)$ $\displaystyle\simeq
2\left[\alpha_{\rm mag}(z_{\rm l})-1\right]\frac{3}{2}H_{0}\Omega_{\rm
m}\int_{0}^{z_{\rm
l}}\frac{\mathrm{d}z~{}H_{0}}{H(z)}\frac{(1+z)^{2}}{1+z_{\rm l}}$
$\displaystyle\times\int\\!{\rm d}z_{\rm s}~{}P_{\rm s}(z_{\rm
s})\frac{\chi^{2}(\chi_{\rm l}-\chi)(\chi_{\rm s}-\chi)}{\chi_{\rm
l}^{2}(\chi_{\rm s}-\chi_{\rm l})}$ $\displaystyle\times\bar{\rho}_{\rm
m0}\int\\!\frac{k\mathrm{d}k}{2\pi}P_{\rm mm}^{\rm
NL}\\!(k;z)J_{2}\left(k\frac{\chi}{\chi_{\rm l}}R\right),$ (4)

where $\alpha_{\rm mag}$ is the power-law slope of the intrinsic number counts
of lens galaxies around the absolute magnitude cut, $P_{\rm s}(z_{\rm s})$ is
the stacked posterior distribution of photometric redshifts for source
galaxies, and $\chi_{\rm l}$ and $\chi_{\rm s}$ are the comoving angular
diameter distances to the lens redshift and the source redshift, respectively.
Note that $\Delta\\!\Sigma^{\rm mag}$ does not depend on the galaxy bias. We
include the redshift distribution of source galaxies, $P_{\rm s}(z_{\rm s})$,
but use the mean redshift of lens galaxies for simplicity. An estimate of the
number counts slope, $\alpha_{\rm mag}$, for each of the absolute magnitude-
limited samples of LOWZ, CMASS1 and CMASS2 is relatively straightforward,
compared to the original parent samples, which have color-dependent flux cuts
that make it challenging to determine the slope for the magnification bias
calculation (von Wietersheim-Kramsta et al., 2021).

From the measured number counts, we estimate $\alpha_{\rm mag}\simeq 2.26\pm
0.03$, $3.56\pm 0.04$ and $3.73\pm 0.04$ for the LOWZ, CMASS1 and CMASS2
sample, respectively, where the $1\sigma$ error is estimated assuming Poisson
errors in the number counts in each magnitude bin around the magnitude cut.
Although we have a relatively accurate estimate of $\alpha_{\rm mag}$ for each
sample, we employ a conservative approach for our cosmological analysis: we
treat $\alpha_{\rm mag}$ as a nuisance parameter in the cosmological analysis,
employing a Gaussian prior with width $\sigma(\alpha_{\rm mag})=0.5$ and a
mean value taken from the above measurement value. As we will show below,
$\Delta\\!\Sigma^{\rm mag}$ constitutes a 1%, 7% and 10% contribution to the
total $\Delta\\!\Sigma^{\rm model}$ for the LOWZ, CMASS1 and CMASS2 sample,
respectively, for the Planck cosmology888Exactly speaking, we should use the
intrinsic redshift distribution of source galaxies for computation of the
magnification bias effect (Eq. 4), rather than the stacked posterior
distribution of photometric redshifts. However, even if we use the
“reweighted” distribution based on the COSMOS catalog, which gives an estimate
of the intrinsic distribution, it changes the magnification bias only by up to
5% fractional changes (therefore by up to $0.5\%(=0.1\times 0.05)$
contamination to the galaxy-galaxy lensing amplitude). In addition, we employ
a broad prior width of $\sigma(\alpha_{\rm mag})=0.5$ as our fiducial choice,
and the $\pm 1\sigma$ shift from the central value in $\alpha_{\rm mag}$ leads
to $\pm 20$–$40\%$ fractional changes in Eq. (4). Hence the prior width in
$\alpha_{\rm mag}$ absorbs the impact of inaccuracy in the model calculation
of Eq. (4) on the cosmological parameters.. Including the
$\Delta\\!\Sigma^{\rm mag}$ contribution in the theoretical template adds some
cosmological information. Encouragingly, we will show that the broad prior on
$\alpha_{\rm mag}$ causes almost no degradation in cosmological parameter
constraints.

The projected auto-correlation function $w_{\rm p}(R)$ is defined in terms of
the three-dimensional correlation function $\xi_{\rm gg}$ as

$\displaystyle w_{\rm p}(R;z_{\rm l})=$ $\displaystyle 2f_{\rm corr}^{\rm
RSD}(R;z_{\rm l})$ $\displaystyle\times\int_{0}^{\pi_{\rm max}}\\!{\rm
d}\Pi~{}\xi_{\rm gg}\left(\sqrt{R^{2}+\Pi^{2}};z_{\rm l}\right),$ (5)

where $\pi_{\rm max}$ is the projection length along the line of sight. We
employ $\pi_{\rm max}=100\,h^{-1}{\rm Mpc}$ as used in the measurement. The
projected correlation function is less sensitive to the RSD effect compared to
$\xi_{\rm gg}$, because the line-of-sight projection reduces the RSD effect,
but the effect is not completely negligible, especially at large $R$. We
account for the residual RSD effect using the Kaiser RSD factor Kaiser (1984):
$f_{\rm corr}^{\rm RSD}(R;z_{\rm l})$ is the correction factor, which depends
on redshift and on the assumed $\Omega_{\rm m}$ and $w_{\rm de}$ for a flat
$w$CDM model (see Eq. (48) in Ref. van den Bosch et al., 2013, for the
definition). Using the minimal bias model, we model the three-dimensional,
real-space galaxy correlation function as

$\displaystyle\xi_{\rm
gg}(r)=b_{1}^{2}\int_{0}^{\infty}\\!\frac{k^{2}\mathrm{d}k}{(2\pi^{2})}~{}P^{\rm
NL}_{\rm mm}(k)j_{0}(kr),$ (6)

where $j_{0}(x)$ is the zeroth-order spherical Bessel function, and $b_{1}$ is
the same bias parameter as in Eq. (2) for each SDSS galaxy sample. While the
galaxy clustering signal is also affected by magnification bias, we have
checked that its contribution is at the sub-percent level compared to Eq. (5),
and hence neglect it in our model.

Throughout this paper, we use the logarithmic center of each separation bin as
a representative projected separation, $R_{i}=(R_{i,{\rm max}}R_{i,{\rm
min}})^{1/2}$, and evaluate the model prediction at the representative
projected separation. We have checked that the difference between the model
signal evaluated at the representative separation and the area-averaged model
signal within each separation bin is below 2% of the square root of the
diagonal element of the covariance matrix.

In this minimal bias model, the relation between the real-space correlation
functions $\xi_{\rm gm}(r)/\sqrt{\xi_{\rm gg}(r)\xi_{\rm mm}(r)}=1$ always
holds by construction. Hence as long as the correlation functions for real
galaxies follow this relation for large separations, our model can accurately
model the measured correlation functions at such separations.

We use Dark Emulator (Nishimichi et al., 2020) to compute the linear matter
power spectrum, $P_{\rm lin}(k)$, for an input cosmological model, which is
constructed from the CLASS Lesgourgues (2011) outputs. We then use the updated
version of halofit (Smith et al., 2008; Takahashi et al., 2012) to compute the
nonlinear matter power spectrum, $P_{\rm NL}(k)$, from the linear power
spectrum for the cosmological model. We use the FFTLog method Hamilton (2000),
implemented in pyfftlog 999A publicly-available python package of FFTLog,
https://github.com/prisae/pyfftlog., to perform the Hankel transforms in Eqs.
(2) and (6). The integration for the galaxy clustering projection in Eq. (5)
is performed with the trapezoidal rule. The model parameters are summarized in
Table 1.

Table 1: Model parameters and priors used in our cosmological parameter inference. The label ${\cal U}(a,b)$ denotes a uniform (or equivalently flat) distribution with minimum $a$ and maximum $b$, while ${\cal N}(\mu,\sigma)$ denotes a normal distribution with mean $\mu$ and width $\sigma$. The parameters above “Extended model” are the parameters used in our baseline analysis: 5 cosmological parameters, a linear galaxy bias parameter and a magnification bias parameter for each of the LOWZ, CMASS1 and CMASS2 samples, and 2 nuisance parameters to model residual photo-$z$ and multiplicative shear errors: $13=5+3+3+2$ in total. We perform the analysis for the “Extended model” after unblinding; we further include the dark energy equation of state parameter, $w_{\rm de}$, and combine the information from $\Delta\\!\Sigma$ and $w_{\rm p}$ with external datasets (e.g. Planck) to estimate all of the cosmological parameters in the extended model, $w$CDM. Parameter | Prior
---|---
Cosmological parameters
$\Omega_{\rm de}$ | ${\cal U}(0.4594,0.9094)$
$\ln(10^{10}A_{\rm s})$ | ${\cal U}(1.0,5.0)$
$\omega_{\rm b}$ | ${\cal N}(0.02268,0.00038)$
$\omega_{\rm c}$ | ${\cal U}(0.0998,0.1398)$
$n_{\rm s}$ | ${\cal N}(0.9649,3\times 0.0042)$
galaxy bias parameters
$b_{1}(z_{i})$ | ${\cal U}(0.1,5.0)$
magnification bias parameters
$\alpha_{\rm mag}(z_{\rm LOWZ})$ | ${\cal N}(2.259,0.5)$
$\alpha_{\rm mag}(z_{\rm CMASS1})$ | ${\cal N}(3.563,0.5)$
$\alpha_{\rm mag}(z_{\rm CMASS2})$ | ${\cal N}(3.729,0.5)$
Photo-$z$/Shear errors
$\Delta z_{\rm ph}$ | ${\cal N}(0.0,0.1)$
$\Delta m_{\gamma}$ | ${\cal N}(0.0,0.01)$
Extended model
after unblinding
$w_{\rm de}$ | ${\cal U}(-4.0,-0.2)$

###  IV.2 Modeling residual systematic errors in the galaxy-galaxy weak
lensing

Miyatake et al. (2021) performed various systematics tests and null tests such
as the $B$-mode signal and the “boost” factor, and did not find any strong
evidence for residual systematic effects in the $\Delta\\!\Sigma$
measurements, reflecting the high quality of the HSC-Y1 data. In our
cosmological analysis, we introduce nuisance parameters $\Delta z_{\rm ph}$
and $\Delta m_{\gamma}$ to model possible residual systematic errors in the
photo-$z$ and multiplicative shear calibration, and adopt conservative priors
on those nuisance parameters. Hence, even if we have residual unknown
systematic effects in the weak lensing measurements, the nuisance parameters
can at least partially absorb their impact on the cosmological constraints. We
assume that the SDSS spectroscopic data has a better control of the systematic
effects, and we do not consider any residual systematic errors in the $w_{\rm
p}$ measurement.

####  IV.2.1 A residual systematic photo-$z$ error: $\Delta z_{\rm ph}$

Photo-$z$ uncertainties are among the most important systematic effects in
weak lensing measurements. To study the impact of photo-$z$ errors on our
results, we introduce a nuisance parameter $\Delta z_{\rm ph}$ to model the
possible residual uncertainty. More specifically, following the method in Ref.
Huterer et al. (2006) (also see Ref. Miyatake et al., 2021, for the detailed
discussion on this modeling of systematics), we model the systematical error
in the mean source redshift by shifting the posterior distribution of each
source galaxy by the same amount, $\Delta z_{\rm ph}$, as

$\displaystyle P_{\rm s}(z_{\rm s})\longrightarrow P_{\rm s}(z_{\rm s}+\Delta
z_{\rm ph}).$ (7)

We then use the shifted distribution to compute the averaged lensing
efficiency $\left\langle\Sigma_{\rm cr}^{-1}\right\rangle_{\rm ls}$ and the
weight $w_{\rm ls}$ for the source-lens pairs using the actual HSC-Y1 and SDSS
catalogs. We found that the lensing signal after this shift can be well
approximated by the following multiplicative form:

$\displaystyle\widehat{\Delta\\!\Sigma}^{(i_{\rm l})}\\!\\!(R;\Delta z_{\rm
ph})\simeq f_{\rm ph}^{(i_{\rm l})}\\!(\Delta z_{\rm
ph})\widehat{\Delta\\!\Sigma}^{(i_{\rm l})}\\!\\!(R;\Delta z_{\rm ph}=0),$ (8)

where $f_{\rm ph}^{(i_{\rm l})}(\Delta z)$ is a multiplicative factor to model
the effect of systematic photo-$z$ error and $i_{\rm l}=$“LOWZ”, “CMASS1” or
“CMASS2” is the index of the lens galaxy sample 101010Here, we implicitly
include the cosmological dependence of $\langle\Sigma_{\rm
cr}^{-1}\rangle_{\rm ls}$ in the photo-$z$ correction, by applying both a
photo-$z$ correction and a measurement correction, i.e., accounting for the
$\Omega_{\rm m}$ and $w_{\rm de}$ dependencies of the measurement described in
Section III, at the same time.. While using a single population of source
galaxies, we find that the shift $\Delta z_{\rm ph}$ leads to different
changes in the amplitudes of $\Delta\\!\Sigma$ for the different lens samples
(LOWZ, CMASS1, and CMASS2) depending on the lens redshift. Conversely, we can
use the relative variations in the $\Delta\\!\Sigma$ amplitudes at different
lens redshifts to calibrate out $\Delta z_{\rm ph}$, simultaneously with
cosmological parameter estimation, if the data has sufficiently high signal-
to-noise ratio. This is a self-calibration method for photo-$z$ error that was
proposed in Oguri and Takada (2011). We will see how effectively the self-
calibration method works to calibrate photo-$z$ errors to higher precision
than the prior, given the statistical power of the HSC-Y1 data and the scale
cuts adopted in this paper.

Exactly speaking, we have to use a shift in the intrinsic redshift
distribution of source galaxies, rather than the posterior distribution, to
estimate the impact of residual photo-$z$ errors on $\Delta\\!\Sigma$. As
discussed in Miyatake et al. (2021), even if we use the intrinsic distribution
estimated by matching the source galaxies to the calibration sample of COSMOS
catalog in magnitude-color space, the effect is very small, compared to the
change caused by using the parameter $\Delta z_{\rm ph}$. Hence we conclude
that our treatment of the residual photo-$z$ error is sufficient to capture
the possible impact.

In our method, we apply the inverse of the photo-$z$ error factor as a
correction to the theoretical template for $\Delta\\!\Sigma(R)$, rather than
changing the measurement, as follows:

$\displaystyle\Delta\\!\Sigma^{\rm
model}(R)\rightarrow\frac{\Delta\\!\Sigma^{\rm model}(R)}{f_{\rm ph}^{(i_{\rm
l})}(\Delta z_{\rm ph})}.$ (9)

This treatment permits us to use an unchanged data vector and its covariance
matrix for the cosmological parameter inference.

Table 2: Differences in the averaged $\left\langle\Sigma_{\rm cl}^{-1}\right\rangle$ using different photo-$z$ method relative to that for the fiducial photo-$z$ method (MLZ). We use the same method given in Eq. (1) to select source galaxies, and compute the average $\left\langle\Sigma_{\rm cl}^{-1}\right\rangle$ over all lens-source pairs in the separation $3\leq R/[h^{-1}{\rm Mpc}]\leq 30$, for each lens sample (LOWZ, CMASS1 and CMASS2). The number after “$\pm$” denotes the $1\sigma$ uncertainty in the difference, estimated from the width of the photo-$z$ posterior of each source galaxy.  photo-$z$ | LOWZ | CMASS1 | CMASS2
---|---|---|---
method |  |  |
DEmP | $-0.048\pm 0.034$ | $-0.000\pm 0.029$ | $-0.030\pm 0.026$
Ephor AB | $-0.046\pm 0.035$ | $0.052\pm 0.033$ | $0.129\pm 0.055$
Franken-Z | $-0.003\pm 0.023$ | $0.002\pm 0.027$ | $0.003\pm 0.030$
Mizuki | $-0.042\pm 0.022$ | $-0.050\pm 0.011$ | $-0.041\pm 0.012$
NNPZ | $-0.049\pm 0.030$ | $-0.018\pm 0.043$ | $0.061\pm 0.050$

As another sanity check, we also study the impact of different photo-$z$
methods on the cosmological results. Table 2 gives differences in
$\left\langle\Sigma_{\rm cr}^{-1}\right\rangle$ when using photo-$z$ estimate
for each source galaxy based on different photo-$z$ methods, relative to that
for the fiducial photo-$z$ method. Note that we repeated the cut defined by
Eq. (1) to define the source galaxy sample for each method, so the source
samples are different for each photo-$z$ method. The different photo-$z$
methods give a few percent change in the $\Delta\\!\Sigma$ amplitudes; these
changes differ depending on the lens samples. We will explicitly study to what
extent the cosmological results are changed by using different photo-$z$
catalogs.

####  IV.2.2 A residual error in multiplicative shear bias: $\Delta
m_{\gamma}$

An accurate weak lensing measurement requires an unbiased measurement of the
shear using ensemble average of shape information for the source galaxies.
Past work has studied the impact of imperfect shape measurements and the
resulting residual systematic uncertainties in the $\Delta\\!\Sigma$
measurements for the HSC-Y1 sample (Mandelbaum et al., 2018b). To model the
impact of possible residual systematic error in the shear calibration, we
introduce a nuisance parameter, $\Delta m_{\gamma}$, and shift the theoretical
template as

$\displaystyle\Delta\\!\Sigma(R)\rightarrow(1+\Delta
m_{\gamma})\Delta\\!\Sigma(R;\Delta m_{\gamma}=0).$ (10)

Then we treat $\Delta m_{\gamma}$ as a nuisance parameter in the cosmological
parameter inference and impose a conservative prior as discussed in Section
IV.3. Since we use a single population of source galaxies as in Oguri and
Takada (2011), we can use the same bias parameter $\Delta m_{\gamma}$ for the
lensing signals for the three lens samples (LOWZ, CMASS1, and CMASS2). This is
a good approximation as long as the source galaxies are well separated from
the lens galaxies. Thus the effect of $\Delta m_{\gamma}$ does not depend on
the lens redshift, so we can distinguish between the two systematic effects of
$\Delta z_{\rm ph}$ and $\Delta m_{\gamma}$, if the signal-to-noise ratio of
the $\Delta\\!\Sigma$ measurement is sufficiently high.

When we include both the photo-$z$ errors and the shear multiplicative errors
in the parameter inference, we simply multiply the model prediction
$\Delta\\!\Sigma$ by the multiplicative functions, $1/f_{\rm ph}(\Delta z_{\rm
ph})$ and $(1+\Delta m_{\gamma})$, assuming that these corrections are
independent.

###  IV.3 Likelihood and parameter estimation

We assume that the likelihood of the HSC-Y1 and SDSS galaxy clustering data
follows a multivariate Gaussian distribution:

$\displaystyle-\ln{\cal L}({\bf
d}|\bm{\theta})=\frac{1}{2}\left[d_{i}-m_{i}(\bm{\theta})\right][{\bf
C}^{-1}]_{ij}\left[d_{j}-m_{j}(\bm{\theta})\right]$ (11)

We employ 14 bins in the range $8\leq R/[h^{-1}{\rm Mpc}]\leq 80$ for $w_{\rm
p}(R)$ and 8 bins in the range $12\leq R/[h^{-1}{\rm Mpc}]\leq 80$ for
$\Delta\\!\Sigma(R)$, respectively. The dimension of the data vector is
$66=3\times(14+8)$ for our baseline analysis setup. We use 13 model
parameters: 5 cosmological parameters, 3 bias parameters and 3 magnification
bias parameters for the three galaxy samples (LOWZ, CMASS1, and CMASS2), and
two nuisance parameters ($\Delta z_{\rm ph}$ and $\Delta m_{\gamma}$). We
adopt the 5 cosmological parameters that specify a flat $\Lambda$CDM model in
our baseline analysis: $\Omega_{\rm de}$ is the present-day density parameter
of the cosmological constant, $\ln(10^{10}A_{\rm s})$ and $n_{\rm s}$ are the
amplitude and tilt parameters of the primordial curvature power spectrum
normalized at $k_{\rm pivot}=0.05~{}{\rm Mpc}^{-1}$, $\omega_{\rm
b}(\equiv\Omega_{\rm b}h^{2})$ is the physical density parameter of baryons,
and $\omega_{\rm c}(\equiv\Omega_{\rm c}h^{2})$ is the physical density
parameter of CDM. Throughout this paper we employ adiabatic initial
conditions.

We use Bayesian parameter inference, where the posterior distribution of the
model parameters with a given data vector is expressed as

$\displaystyle{\cal P}(\bm{\theta}|{\bf d})\propto{\cal L}({\bf
d}|\bm{\theta})\Pi(\bm{\theta}),$ (12)

where $\Pi(\bm{\theta})$ is the prior on the parameters. The choice of prior
is summarized in Table 1. Since the clustering observables $\Delta\\!\Sigma$
and $w_{\rm p}$ are not sensitive to $\omega_{\rm b}$ and $n_{\rm s}$, we
employ priors on those parameters from other experiments. For the prior on
$\omega_{\rm b}$ we employ a normal distribution with mean and width inferred
from Big Bang nucleosynthesis (BBN) experiments (Aghanim et al., 2020; Aver et
al., 2015; Cooke et al., 2018; Schöneberg et al., 2019). For the prior $n_{\rm
s}$ we employ a normal distribution given in Table 1, where we adopt the
Planck 2018 result (see “TT,TE,EE+lowE+lensing” column of Table 2 in Ref.
Aghanim et al., 2020) for the mean and a width three times wider than the
Planck error as a conservative choice. For the residual photo-$z$ error
parameter, we employ a normal distribution with zero mean and width
$\sigma(\Delta z_{\rm ph})=0.1$, which is a conservative choice and larger
than the uncertainty inferred from the photo-$z$ method (a few per cent) (see
Table. 5 in Hikage et al., 2019). For the residual shear error parameter, we
employ a normal distribution prior with zero mean and width $\sigma(\Delta
m_{\gamma})=0.01$, which is estimated from HSC galaxy image simulations
Mandelbaum et al. (2018b). For other parameters we adopt a uniform
distribution with sufficiently wide width so that these priors are not
informative for the cosmological parameter constraints.

We sample the posterior distribution in a multi-dimensional parameter space by
using the nested sampling technique implemented in MultiNest (Feroz and
Hobson, 2008; Feroz et al., 2009, 2019) through the interface for cosmological
parameter inference MontePython (Audren et al., 2013; Brinckmann and
Lesgourgues, 2019). We set the following MultiNest hyper parameters: the
sampling efficiency parameter efr=0.8, the evidence tolerance factor tol=0.5,
and number of live points Nlive=1000.

After sampling the posterior distribution, we estimate the central value and
credible interval for the cosmological parameter(s) from the chain. In this
paper, we report the mode as the central value and the 68% highest density
interval as the credible interval. The definitions of the mode and the highest
density interval are illustrated in Fig. 3. The mode is defined by the
parameter value that has the highest posterior probability, and the 68%
highest density interval is defined so that the probability integrated within
the interval is 68%. To obtain the mode and the highest density interval from
a given chain, we first estimate the marginalized posterior distribution by
using kernel density estimation using the public code getdist (Lewis, 2019),
and then find the mode and the highest density interval from the posterior
distribution.

Among the sampled model parameters, we focus on $\Omega_{\rm m}$,
$\sigma_{8}$, and $S_{8}\equiv\sigma_{8}(\Omega_{\rm m}/0.3)^{0.5}$, which are
derived from the sampled parameters. In Appendix B, we show the convergence of
our sample chains and that our setup is suitable to infer the cosmological
parameters to within a sub-percent level of the 68% credible interval.

Figure 3: An illustration of the definitions of the mode and 68% highest
density interval. The black line is the 1-d marginalized posterior
distribution, ${\cal P}(\theta)$, of a model parameter $\theta$. The mode is
defined as the parameter value which has highest probability. The 68% highest
density interval is a interval in which the probability is higher than outside
and the total probability is 68%, where probability density is normalized so
that $\int_{0}^{\infty}{\rm d}\theta{\cal P}(\theta)$=1.

##  V Blinding scheme and Validation

In this section, we describe our strategy of analysis to secure the robustness
of the results to confirmation bias and systematics in the data, analysis and
model.

To avoid confirmation bias we perform our cosmological analysis in a blind
fashion. The details are described in Section 3.2 of Hikage et al. (2019). We
employ a two-tier blinding strategy to avoid accidental unblinding during the
cosmological analysis; namely, both at the catalog level and at the analysis
level:

  * •

catalog level – the analysis team performs the full cosmological analysis
using three different catalogs of galaxy shapes for weak lensing measurements,
one of which is the true catalog and the other two are fake (see below for
details). No member in the analysis team knows which catalog is the true
catalog.

  * •

analysis level – the analysis team is not allowed to make a plot comparing the
measurement with theoretical models. When the analysis team makes a plot
showing the credible regions of cosmological parameters (i.e. the posterior
distribution), the central value(s) of parameter(s) are shifted to zero, and
only the shifted range of the credible region(s) can be seen.

  * •

analysis level – The analysis team is not allowed to compare the posterior of
cosmological parameter(s) or the inferred model predictions with external
results such as the Planck CMB cosmology.

Use of the three shape catalogs means that the analysis group must perform
three analyses, but this method avoids the need for reanalyses once the
catalogs are unblinded.

In addition we did not make any comparison between the posterior distributions
of parameters obtained from this paper and from the companion paper using the
halo model method, Miyatake et al. (2021), during the blind analysis stage.
Our cosmological analysis method has been tested and demonstrated to be valid
in Sugiyama et al. (2020).

In addition to our baseline analysis, we employ alternative analysis setups to
quantify internal consistency and systematic effects in parameter estimation
before unblinding, as summarized in Table 3.

Table 3: The list of our analysis setups: the baseline analysis setup and those designed to check for internal consistency. The marks “$\checkmark$” or “–” in each column denote whether each analysis does or does not include the parameter(s) in the inference. The column labelled as ${\bm{\theta}}_{\rm cosmo}$ denotes the set of cosmological parameters, $\bm{\theta}_{\rm cosmo}=\\{\Omega_{\rm de},\ln(10^{10}A_{\rm s}),n_{\rm s},\omega_{\rm c},\omega_{\rm b}\\}$. The other parameters are the same as in Table 1.  setup label | sample parameter | comment
---|---|---
| $\bm{\theta}_{\rm cosmo}$ | $(b_{1,i})$ | $\Delta m_{\gamma}$ | $\Delta z_{\rm ph}$ | $\alpha_{\rm mag}$ | $w$ |
baseline | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | – | baseline analysis: $\bm{\theta}_{\rm cosmo}=\\{\Omega_{\rm de},\ln(10^{10}A_{\rm s}),n_{\rm s},\omega_{\rm c},\omega_{\rm b}\\}$
w/o LOWZ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | – | w/o the LOWZ sample
w/o CMASS1 | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | – | w/o the CMASS1 sample
w/o CMASS2 | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | – | w/o the CMASS2 sample
w/o photo-$z$ error | $\checkmark$ | $\checkmark$ | $\checkmark$ | – | $\checkmark$ | – | fixing $\Delta z_{\rm ph}=0$
w/o shear calib. error | $\checkmark$ | $\checkmark$ | – | $\checkmark$ | $\checkmark$ | – | fixing $\Delta m_{\gamma}=0$
w/o mag. effect error | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | – | – | fixing $\alpha_{\rm mag}=0$
2 cosmo. paras | $\triangle$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | – | fixing $(n_{\rm s},\omega_{\rm b},\omega_{\rm c})$ to the Planck best-fit values: $\bm{\theta}_{\rm cosmo}=\\{\Omega_{\rm de},\ln(10^{10}A_{\rm s})\\}$
$w$CDM | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | including the DE equation of state parameter, $w_{\rm de}$, after unblinding

##  VI Results

###  VI.1 Cosmological parameters for the $\Lambda$CDM model

In this section, we present the results of cosmological constraints in a flat
$\Lambda$CDM model. All the analysis we show in this section were done before
unblinding 111111After unblinding, we found a bug in the analysis pipeline
that the width of the Gaussian prior was mistakenly set to a value smaller
than the correct value by a factor of $2^{1/2}$. The results in this section
were from the reanalyses with this bug fixed, done after unblinding. However,
we believe that the blinding strategy was still effective and our results are
free from confirmation bias, because the results before and after the bug
fixed were almost unchanged; therefore we concluded that the unblinding
criteria are not affected.. We neither saw the actual values of the
cosmological parameters, nor compared our results with those from any other
analysis, including those from Planck or our companion paper, Miyatake et al.
(2021).

Table 4: Summary of cosmological constraints for different analysis setups (see Table 3). Here we report the mode of the marginalized posterior along with the credible interval defined by the highest density interval (see Section IV.3 and Fig. 3). setup | $S_{8}\equiv\sigma_{8}(\Omega_{\mathrm{m}}/0.3)^{0.5}$ | $\sigma_{8}$ | $\Omega_{\rm m}$ | $w_{0}$
---|---|---|---|---
baseline | $0.936^{+0.092}_{-0.086}$ | $0.85^{+0.16}_{-0.11}$ | $0.283^{+0.12}_{-0.035}$ | $-$
w/o LOWZ | $1.13^{+0.14}_{-0.13}$ | $1.02^{+0.17}_{-0.15}$ | $0.328^{+0.12}_{-0.054}$ | $-$
w/o CMASS1 | $0.99^{+0.10}_{-0.12}$ | $0.94^{+0.19}_{-0.15}$ | $0.255^{+0.11}_{-0.037}$ | $-$
w/o CMASS2 | $0.92^{+0.11}_{-0.12}$ | $0.82^{+0.16}_{-0.14}$ | $0.319^{+0.10}_{-0.052}$ | $-$
w/o multiplicative bias | $0.932^{+0.095}_{-0.089}$ | $0.84^{+0.17}_{-0.11}$ | $0.288^{+0.12}_{-0.037}$ | $-$
w/o photo-$z$ | $0.941^{+0.078}_{-0.082}$ | $0.85^{+0.15}_{-0.10}$ | $0.298^{+0.11}_{-0.042}$ | $-$
w/o mag. effect | $0.939^{+0.088}_{-0.087}$ | $0.86^{+0.14}_{-0.12}$ | $0.287^{+0.12}_{-0.034}$ | $-$
DEmP | $0.931^{+0.083}_{-0.089}$ | $0.85^{+0.18}_{-0.11}$ | $0.270^{+0.11}_{-0.035}$ | $-$
Ephor AB | $1.071^{+0.077}_{-0.082}$ | $1.07^{+0.14}_{-0.16}$ | $0.260^{+0.090}_{-0.041}$ | $-$
Franken-Z | $0.948^{+0.086}_{-0.087}$ | $0.86^{+0.17}_{-0.11}$ | $0.278^{+0.11}_{-0.037}$ | $-$
Mizuki | $0.892^{+0.089}_{-0.085}$ | $0.79^{+0.17}_{-0.10}$ | $0.28^{+0.13}_{-0.030}$ | $-$
NNPZ | $0.952^{+0.092}_{-0.088}$ | $0.86^{+0.17}_{-0.11}$ | $0.288^{+0.12}_{-0.036}$ | $-$
2 cosmo | $0.939^{+0.090}_{-0.087}$ | $0.909^{+0.10}_{-0.089}$ | $0.311^{+0.036}_{-0.037}$ | $-$
$w$CDM (HSC$\times$Planck2018) | $0.817^{+0.022}_{-0.021}$ | $0.892^{+0.051}_{-0.056}$ | $0.246^{+0.045}_{-0.035}$ | $-1.28^{+0.20}_{-0.19}$
Figure 4: The 1-d and 2-d posterior distributions of the three main
cosmological parameters, $\Omega_{\rm m}$, $\sigma_{8}$ and $S_{8}$, obtained
from the baseline analysis (see Table 3), where we use the galaxy clustering
signals over $8<R/[h^{-1}{\rm Mpc}]<80$ and the galaxy-galaxy lensing signals
over $12<R/[h^{-1}{\rm Mpc}]<80$. The dark (light) shaded region shows the 68%
(95%) credible interval. The posterior distributions over the full
13-parameter space are shown in Fig. 16. Figure 5: Similar to the previous
figure, now showing the 1-d and 2-d posterior distributions for $S_{8}$ and
the nuisance parameters $\Delta m_{\gamma}$ (multiplicative shear error) and
$\Delta z_{\rm ph}$ (photo-$z$ bias) for the HSC source galaxies used in the
weak lensing measurements. The vertical dashed lines in the 1-d distributions
of $\Delta m_{\gamma}$ and $\Delta z_{\rm ph}$ denote the width of the
Gaussian prior of these parameters.

Fig. 4 shows the posterior distributions of $S_{8}$, $\sigma_{8}$ and
$\Omega_{\rm m}$ for the baseline analysis. The central value and the credible
interval of each parameter are, respectively:

$\displaystyle S_{8}$ $\displaystyle=0.936^{+0.092}_{-0.086}$ (13)
$\displaystyle\sigma_{8}$ $\displaystyle=0.85^{+0.16}_{-0.11}$
$\displaystyle\Omega_{\rm m}$ $\displaystyle=0.283^{+0.12}_{-0.035}$

Thus the joint measurements of $\Delta\\!\Sigma$ and $w_{\rm p}$ from the
HSC-Y1 and SDSS catalogs achieve a precision of $\sigma(S_{8})\simeq 0.09$.
This precision can be considered as a conservative and robust constraint,
because we demonstrated in Sugiyama et al. (2020) that any systematic bias in
$S_{8}$ due to inaccuracies in the minimal bias model is very unlikely to
become larger than the 68% credible interval. However, our result displays a
slightly larger value of $S_{8}$ compared to other results such as the Planck
result, as we below discuss in more detail. For completeness, we show the
posterior distributions of all 13 parameters (and several derived parameters)
in Appendix C. A closer look at Fig. 4 shows that the posterior distribution
of $\Omega_{\rm m}$ has a flat-shaped peak. We found that this arises when we
combine the cosmological information from the three galaxy samples, while the
cosmological parameters from each of the galaxy samples alone differ due to
sample variance, as shown in Appendix D.1 (also see Ivanov et al., 2020). The
flat-shaped peak is also partially a consequence of the degeneracy between
$\Omega_{\rm m}$ and $\omega_{\rm c}$. If we fix $\omega_{\rm c}$ to the
Planck 2015 best-fit value, $\Omega_{m}$ shows a narrower and peakier
distribution.

As discussed in detail in Sugiyama et al. (2020), using either of
$\Delta\\!\Sigma(R)$ or $w_{\rm p}(R)$ alone cannot constrain these parameters
simultaneously and suffers from severe degeneracies. On large scales where
linear theory holds, $\Delta\\!\Sigma(R)$ is proportional to
$b_{1}\sigma_{8}^{2}$, while $w_{\rm p}(R)$ is proportional to
$b_{1}^{2}\sigma_{8}^{2}$. Thus either alone cannot constrain $b_{1}$ and
$\sigma_{8}$ separately. If $\sigma_{8}$ is very large, that assumption is
incorrect, and our model will overpredict $\Delta\\!\Sigma$ and $w_{\rm p}$
around the minimum scale because the stronger nonlinear effect boosts the
amplitudes around the scale compared to what the linear theory predicts.
Because of this, a model with extremely large $\sigma_{8}$ is disfavored in
parameter inference, while there is no such penalty for arbitrarily small
values of $\sigma_{8}$. Thus the resultant posterior distribution with either
$\Delta\\!\Sigma$ or $w_{\rm p}$ alone depends on the lower limit of the prior
range of $\sigma_{8}$. Hence only the joint analysis of $\Delta\\!\Sigma$ and
$w_{\rm p}$ gives meaningful constraints on $\sigma_{8}$ and $S_{8}$ (see also
Fig. 7 of Ref. Sugiyama et al., 2020). As shown in Appendix C, the bias
parameter for each galaxy sample is determined to a fractional precision of
about 20%. $\Omega_{\rm m}$ is constrained relatively well because a change in
$\Omega_{\rm m}$ causes a scale-dependent modification in $w_{\rm p}$ (and
$\Delta\\!\Sigma$), so the $w_{\rm p}$ information in our datasets can give a
meaningful constraint on the parameters.

Fig. 5 shows the posterior distributions of $S_{8}$ with the nuisance
parameters characterizing photo-$z$ errors and multiplicative shear bias. The
posterior distributions of the nuisance parameters are prior-dominated (see
Table 1), meaning that $\Delta\\!\Sigma(R)$ and $w_{\rm p}(R)$ on large scales
alone cannot constrain these parameters well. As shown in Miyatake et al.
(2021) the inclusion of smaller-scale information in the analysis constrains
the photo-$z$ parameter better than the prior range. In addition, Miyatake et
al. (2021) presented the cosmological parameters when employing an even
broader width of $\sigma(\Delta z_{\rm ph})=0.2$, instead of $\sigma(\Delta
z_{\rm ph})=0.1$ in the fiducial prior, and showed that the data allows for a
self-calibration of the residual photo-$z$ bias, indicating $\Delta z_{\rm
ph}=-0.113$ for the central value, as can be found from Fig. 26 of their
paper. As the post-unblinding analysis, we found that $S_{8}$ is changed to
$S_{8}=0.902^{+0.077}_{-0.082}$ from $S_{8}=0.936^{+0.092}_{-0.086}$, if we
adopt the fixed value of $\Delta z_{\rm ph}=-0.113$ in the parameter
inference. Thus the larger value of $S_{8}$ in our results might be partly
from the possible residual photo-$z$ bias. However, the shift is not
significant, and we need more HSC data to obtain a more definite conclusion.

Figure 6: An evaluation of goodness-of-fit of the best-fit model at maximum a
posteriori (MAP) for the baseline analysis. The magenta histogram shows the
distribution of the $\chi^{2}$ value of the model at MAP, obtained by applying
the same baseline analysis to each of 30 noisy mock datasets (see text for
details). The magenta line is the best-fit $\chi^{2}$ distribution,
characterized by the degrees of freedom $\nu=62.3$ estimated from the same
mocks. The vertical black line indicates $\chi^{2}=69.1$ at MAP for the real
analysis of the HSC-Y1 and SDSS data. The blue dashed line shows the
$\chi^{2}$ distribution with degrees of freedom $\nu=60.5$ computed using the
Gaussian linear model (GLM) in Raveri et al. (2020) (see text for details).

Fig. 6 shows an evaluation of goodness-of-fit of the best-fit model, evaluated
at maximum a posteriori (MAP) of the posterior distribution, for the baseline
analysis. To make the reference $\chi^{2}$ distribution, we need to evaluate
the effective degrees of freedom, which generally differ from the naive
evaluation of degrees of freedom, i.e. $\nu=\nu_{\rm data}-\nu_{\rm
param}=66-13=53$ in our case, because of the parameter degeneracies. To obtain
the effective degrees of freedom, we first generate noisy mock data vectors of
$\Delta\\!\Sigma(R)$ and $w_{\rm p}(R)$ for the galaxy samples by adding
random statistical scatter to the noiseless mock signals of
$\Delta\\!\Sigma(R)$ and $w_{\rm p}(R)$. Note that the noiseless mock signals
were generated using $N$-body simulation data for the Planck 2015 cosmology
Sugiyama et al. (2020) (also see Nishimichi et al., 2019). The histogram in
the figure shows the distribution of $\chi^{2}$ for the model at MAP for each
of the 30 noisy mocks. The magenta line denotes the best-fit
$\chi^{2}$-distribution, specified by the effective degrees of freedom
$\nu_{\rm eff}=62.3$. The actual $\chi^{2}$ at MAP for the real analysis of
HSC-Y1 and SDSS data is $\chi^{2}=69.1$, corresponding to a $p$-value of
$0.259$, which indicates that the minimal bias model gives an acceptable fit
to the data within the error bars. For a further comparison, we also compute
the expected $\chi^{2}$ distribution following the “Gaussian linear model”
(GLM) in Ref. Raveri et al. (2020), which assumes that both data vector and
model parameters are Gaussian-distributed. Here we generate 10,000 noisy mock
data vectors around the above mock signals, compute the MAP model for each of
them, and compute $\chi^{2}$ at MAP using the GLM method. The blue dashed line
gives the best-fit $\chi^{2}$-distribution, specified by the effective degrees
of freedom $\nu_{\rm eff}=60.5$ estimated from these 10,000 mocks. The GLM
method gives a consistent $\chi^{2}$ distribution with that estimated from the
analysis using the noisy mock signals. Hence we conclude the MAP model gives
an acceptable fit to the measured signals.

Figure 7: Summary of the constraints on cosmological parameters for the
different setups (see Table 3). Here the dot symbol and error bar in each
column denote the central value and the 68% credible interval (Fig. 3). For
comparison, the shaded band shows the 68% credible interval of the baseline
analysis.

Fig. 7 summarizes the 1-d posterior distribution of $S_{8}$, $\sigma_{8}$ and
$\Omega_{\rm m}$ for the different setups in Table 3. We do not identify any
significant shift or sign of systematic effects in the parameter estimation;
in particular, all the $S_{8}$ results are consistent with the baseline setup
to within the 68% credible interval. In the following we discuss each of the
results.

Figure 8: Similar to Fig. 4, now showing the posterior distributions when we
remove one of the galaxy samples (LOWZ, CMASS1 or CMASS2) from the parameter
inference.  Figure 9: Similar to Fig. 4, now showing the posterior
distributions for each galaxy sample alone. Figure 10: Similar to Fig. 4, now
showing the results when we use different photo-$z$ catalogs used to define
the HSC source galaxies for the $\Delta\\!\Sigma$ measurements (see text for
details).

Fig. 8 shows the results if we remove one of the three galaxy samples (LOWZ,
CMASS1 or CMASS2) from the parameter inference. In Fig. 9 we show the
posterior distributions for each galaxy sample separately. In both cases, the
results are consistent with one another. These plots demonstrate that the
scatter in the posterior distribution has an appreciable contribution from
sample variance.

In Fig. 10 we study how the inferred cosmological parameters are changed if we
use the different photo-$z$ catalogs of HSC source galaxies based on Eq. (1)
for the $\Delta\\!\Sigma$ measurements. We used the same covariance matrix as
for the fiducial photo-$z$ catalog for these analyses, because the effect of
different photo-$z$ methods on the covariance matrix was shown to be small by
Ref. Shirasaki et al. (2019). Keeping the covariance matrix fixed also allows
a more direct comparison of the sensitivity of cosmological parameters to the
choice of photo-$z$ catalog. We find consistent results within the credible
intervals for all photo-$z$ samples except for “Ephor AB”, which differs
significantly from the results of the baseline analysis. The Ephor-AB sample
has a substantially smaller number of source-lens pairs, giving a noisier
$\Delta\\!\Sigma$ signal. We explore this further in Appendix D.2, where we
find that the difference in $\sigma_{8}$ and $S_{8}$ is caused largely by the
CMASS2 signal at large separations, $R>10h^{-1}{\rm Mpc}$; if we remove the
CMASS2 sample, we find much more consistent results.

Figure 11: The posterior distributions when using only the two cosmological
parameters, $\Omega_{\rm de}$ and $\ln(10^{10}A_{\rm s})$, in the parameter
inference, instead of the five parameters in the baseline analysis in Fig. 4.
We fixed the values of the other three cosmological parameters to those
inferred from the Planck 2015 “TT,TE,EE+lowP” constraints, and we included the
same nuisance parameters as in the baseline analysis. The $S_{8}$ result is
almost unchanged.

Fig. 11 shows the results when allowing only the two cosmological parameters,
$\Omega_{\rm de}$ and $\ln(10^{10}A_{\rm s})$ to vary, while keeping the other
three parameters, $(\omega_{\rm b},\omega_{\rm c},n_{\rm s})$ to the best-fit
values of the Planck 2015 “TT,TE,EE+lowP” constraints (Aghanim et al., 2020).
Encouragingly the $S_{8}$ result is almost unchanged, confirming that $S_{8}$
is close to the principal parameter that the clustering observables can most
accurately constrain. In n other words, other three parameters do not strongly
affect the inference from clustering observables. On the other hand, the
$\Omega_{\rm m}$ constraint is considerably weaker when allowing all five
parameters to vary, because of the strong degeneracy between $\Omega_{\rm m}$
and $\omega_{\rm c}$ in a flat $\Lambda$CDM model. Since $\sigma_{8}$ and
$\Omega_{\rm m}$ are correlated in our analysis (Fig. 16), this leads to a
degradation in the constraints on $\sigma_{8}$.

###  VI.2 Cosmological results of post-unblinded analyses

In this section, we show the results of post-unblinding analyses.

Figure 12: Comparison of our constraints (“HSC-Y1 GCC$\times$GGL, minimal
bias”) with other cosmological experiments for flat-geometry $\Lambda$CDM
cosmologies. For the galaxy survey constraints, we focus on the results that
are obtained from clustering information using a similar setup to ours: “GGC”
denotes the projected galaxy clustering, “GGL” denotes the galaxy-galaxy weak
lensing, and “CS” denotes the cosmic shear information. Magenta contours show
the DES-Y1 results Abbott et al. (2018), while brown contours show the
KiDS-1000 result (Heymans et al., 2021). The orange contours show the results
obtained in our companion paper, Miyatake et al. (2021), when including
smaller scale information with the HOD based method for the same HSC-Y1 data,
“HSC-Y1 GCC$\times$GGL, Miyatake HOD”. The green contours show the results for
cosmic shear tomography, “HSC-Y1 GC, Hikage” (Hikage et al., 2019). The cyan
contours show the Planck CMB constraint.

In Fig. 12, we compare cosmological parameter estimation results with those
from other experiments for the flat $\Lambda$CDM model. Throughout, we assumed
a fixed value of total neutrino mass, $m_{\nu,{\rm tot}}=0.06\,$eV. For the
Planck result, we consider the Planck 2018 (Aghanim et al., 2020) cosmological
constraints, in particular those derived from primary CMB information,
referred to as “TT, EE, TE+lowE” in their paper. In other words, we do not
include CMB lensing information. We used the publicly available likelihood
code for the Planck data 121212https://wiki.cosmos.esa.int/planck-legacy-
archive/index.php/Cosmological_Parameters to infer the cosmological parameters
when the neutrino mass is fiexd to 0.06 eV. The HSC-Y1 cosmic shear result
(Hikage et al., 2019) is taken from the website131313https://hsc-
release.mtk.nao.ac.jp/archive/filetree/s16a-shape-
catalog/pdr1_hscwl/Hikage/HSC_Y1_LCDM_post_mnu0.06eV.txt. For the other
lensing experiments that we compare to, we use the cosmological constraints
that are obtained from lensing data in a similar setup to what we employ in
this paper. For the DES-Y1 Abbott et al. (2018) and KiDS-1000 (Heymans et al.,
2021) results, we use cosmological constraints from a combination of galaxy-
galaxy lensing (“GGL”), galaxy clustering (“GGC”) and cosmic shear (“CS”).
These lensing surveys cover a wider area than does the HSC-Y1 ($\sim 1000$
deg2 compared to $140$ deg2), but are shallower than is HSC-Y1. We used public
results, available from the websites 141414http://desdr-
server.ncsa.illinois.edu/despublic/y1a1_files/chains/wg_l3.txt and
151515http://kids.strw.leidenuniv.nl/DR4/data_files/KiDS1000_3x2pt_fiducial_chains.tar.gz,
for DES-Y1 and KiDS-1000, respectively. We note that the DES-Y1 result was
obtained by varying the neutrino mass. However, this difference is not
important because the neutrino mass is not well constrained by lensing
information or large-scale structure information alone. We refer to the
“CS”$\times$“GGL” result for KiDS-1000, as that KiDS-1000 analysis used
redshift-space galaxy clustering information, and included the BAO
information. Hence, Fig. 12 tries to give an apple-to-apple comparison between
the lensing results.

Our cosmological results are weaker than the other constraints we show,
because our method only uses large-scale clustering information in the quasi-
nonlinear regime where the minimal bias model is valid. In our companion paper
(Miyatake et al., 2021), we incorporate smaller-scale data using an HOD model.
As shown in Fig. 12, the resulting constraints, denoted as “HSC-Y1
GGC$\times$GGL, Miyatake HOD”, become comparable with the results of other
lensing experiments, despite the fact that the solid angle covered by HSC-Y1
is 10 times smaller than that of DES-Y1 or KiDS-1000. Hence our constraints
can be considered as a conservative result. In addition, our result is
consistent with the results of both Planck 2018 and the HSC-Y1 cosmic shear
analysis.

Figure 13: The posterior distributions of parameters including the dark energy
of equation parameter, $w_{\rm de}$, for the joint parameter inference of the
HSC-Y1, SDSS and Planck2018 data for flat-geometry $w$CDM cosmologies. For
Planck, we include the primary anisotropy data for temperature and $E$-mode
polarization, denoted as “TT,TE,EE+lowE” (see Aghanim et al., 2020). That is,
we did not include the CMB lensing information.

In Fig. 13, we show the cosmological parameters estimated from the joint
analysis of our HSC-Y1 likelihood and the Planck 2018 likelihood for a flat
$w$CDM model. Here we add the dark energy equation of state parameter, $w_{\rm
de}$, in addition to parameters we have so far used for the $\Lambda$CDM
model. We employ the priors described in Section IV.3 and Table 1 except for
$\omega_{\rm b}$ and $n_{\rm s}$, for which we employ uniform priors ${\cal
U}(0.02190,0.02285)$ and ${\cal U}(0.9500,0.9781)$, respectively. We use the
public Planck 2018 likelihood code, for the primary anisotropy information
(“TT,EE,TE+lowE”), to perform the joint analysis. Since the Planck information
strongly constrains some of the cosmological parameters such as $\omega_{\rm
b}$, $\omega_{\rm c}$ and $n_{s}$, the joint analysis helps breaking parameter
degeneracies. Nevertheless, we note that the Planck information alone cannot
constrain $w_{\rm de}$: the posterior distribution of $w_{\rm de}$ for Planck
alone extends to the lower edge of the prior range. We also note that the
addition of $w_{\rm de}$ causes strong degeneracies between the cosmological
parameters, reflecting that the Planck information alone cannot constrain
these parameters simultaneously. Hence we should not consider seriously about
the consistency between the Planck result and our result. On the other hand,
when the HSC constraints in the local universe is combined with the Planck
constraints, it allows us to infer the growth of large-scale structure and
then use it to constrain the equation of state parameter of dark energy,
$w_{\rm de}$. The joint analysis now shows tightened constraints on $S_{8}$,
$\Omega_{\rm m}$ and $\sigma_{8}$ (see Table 4). The joint analysis prefers a
value of $w_{\rm de}$ slightly smaller than $-1$, but the deviation from this
value is not significant (also see Fig. 14), meaning that the inferred model
is consistent with a flat $\Lambda$CDM model.

Figure 14: Similar to the previous figure, now showing a zoom-in version of
the posterior distribution in the sub-space of $(w_{\rm de},S_{8})$.

##  VII Summary

In this paper we have presented cosmological constraints from a joint-probe
cosmology analysis combining the galaxy-galaxy weak lensing and the projected
correlation function, measured from the HSC-Y1 imaging galaxy catalog and the
spectroscopic SDSS galaxy catalog. To do this, we adopted a conservative
standing point: we employed a perturbation theory based model, more
specifically the “minimal bias” model, as the theoretical template to
interpret the clustering observables. As shown in our validation paper
(Sugiyama et al., 2020), this method can properly extract the cosmological
parameters in an unbiased way as long as the analysis is restricted to large
scales ($R\gtrsim 10\,h^{-1}{\rm Mpc}$), as nonlinear effects such as
nonlinear clustering and baryonic physics are confined to local scales. This
method is also robust against the complication of assembly bias, because
assembly bias changes the clustering amplitudes on large scales in such a way
that the cross-correlation coefficient function $r_{\rm gg}(r)\equiv\xi_{\rm
gm}(r)/[\xi_{\rm gg}(r)\xi_{\rm mm}(r)]^{1/2}$ is close to unity (Hadzhiyska
et al., 2021), which is satisfied by the minimal bias model by definition.

For our baseline analysis we employed the BBN prior on $\omega_{\rm b}$ and
the Planck prior on $n_{\rm s}$, but adopted broad priors on other parameters
including the linear galaxy bias parameters, $b_{1}$, for the three galaxy
samples. The cosmological constraints we obtained are summarized as
$S_{8}=0.936^{+0.092}_{-0.086}$, $\sigma_{8}=0.85^{+0.16}_{-0.11}$, and
$\Omega_{\rm m}=0.283^{+0.12}_{-0.035}$ for the flat $\Lambda$CDM model. Thus,
encouragingly, the joint-probe cosmology helps lift degeneracies between the
cosmological parameters and the bias parameters; the bias parameters are
constrained to a fractional precision of about 20% for each galaxy sample.
However, we found a non-negligible degeneracy between $\omega_{\rm c}$ and the
aforementioned parameters. We further combined the HSC-Y1 and SDSS constraints
with the Planck likelihood, and found $S_{8}=0.817^{+0.022}_{-0.021}$,
$\sigma_{8}=0.892^{+0.051}_{-0.056}$, $\Omega_{\rm m}=0.246^{+0.045}_{-0.035}$
and $w_{\rm de}=-1.28^{+0.20}_{-0.19}$ for the flat $w$CDM model. These
parameters are consistent with the flat $\Lambda$CDM model inferred from the
Planck 2018 experiment. The statistical errors on the parameters from our
analysis are still significantly larger than those which include small-scale
information (Miyatake et al. 2021). However, we believe that the true
cosmological model should be contained within the credible regions of our
favored models, if the $\Lambda$CDM framework is correct. In our analysis we
found no evidence for residual systematic errors comparable to the current
statistical errors.

We can expect further improvements in the cosmological constraints with the
upcoming HSC data. The galaxy shape catalog of the HSC Year 3 data, covering
450 deg2 (3 times that of HSC-Y1) and described in Ref. Li et al. (2021),
should considerably improve the statistical precision of the weak lensing
measurements. We have now developed the cosmology analysis pipeline, and we
will use it on a HSC Year 3 analysis. We will carry out a 3$\times$2pt
analysis (Abbott et al., 2018), combining the HSC and SDSS joint-probe
analysis with the HSC cosmic shear information.

###### Acknowledgements.

We would like to thank Rachel Mandelbaum for very useful comments during this
work was done. This work was supported in part by World Premier International
Research Center Initiative (WPI Initiative), MEXT, Japan, and JSPS KAKENHI
Grant Numbers JP15H03654, JP15H05887, JP15H05893, JP15H05896, JP15K21733,
JP17H01131, JP17K14273, JP18H04350, JP18H04358, JP19H00677, JP19K14767,
JP20H00181, JP20H01932, JP20H04723, JP20H05850, JP20H05855, JP20H05861,
JP21J00011, JP20H05856, JP21H01081 and JP21J10314 by Japan Science and
Technology Agency (JST) CREST JPMHCR1414, by JST AIP Acceleration Research
Grant Number JP20317829, Japan, and by Basic Research Grant (Super AI) of
Institute for AI and Beyond of the University of Tokyo. SS is supported by
International Graduate Program for Excellence in Earth-Space Science (IGPEES),
World-leading Innovative Graduate Study (WINGS) Program, the University of
Tokyo. HM and MSi were supported by the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and
Space Administration. KO is supported by JSPS Research Fellowships for Young
Scientists. YK was supported by the Advanced Leading Graduate Course for
Photon Science at the University of Tokyo. The Hyper Suprime-Cam (HSC)
collaboration includes the astronomical communities of Japan and Taiwan, and
Princeton University. The HSC instrumentation and software were developed by
the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for
the Physics and Mathematics of the Universe (Kavli IPMU), the University of
Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia
Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and
Princeton University. Funding was contributed by the FIRST program from
Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science
and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS),
Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ,
Kavli IPMU, KEK, ASIAA, and Princeton University. This paper makes use of
software developed for the Vera C. Rubin LSST. We thank the LSST Project for
making their code available as free software at http://dm.lsst.org. The Pan-
STARRS1 Surveys (PS1) have been made possible through contributions of the
Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project
Office, the Max-Planck Society and its participating institutes, the Max
Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for
Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham
University, the University of Edinburgh, Queen’s University Belfast, the
Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory
Global Telescope Network Incorporated, the National Central University of
Taiwan, the Space Telescope Science Institute, the National Aeronautics and
Space Administration under Grant No. NNX08AR22G issued through the Planetary
Science Division of the NASA Science Mission.

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##  Appendix A Magnification bias effect

In this appendix, we derive the additional contribution due to magnification
bias to the covariance matrix of the lensing profile $\Delta\\!\Sigma$.

The lensing magnification due to large-scale structure between us and lens
galaxies causes modulations in the number densities of lens galaxies. As a
result, the observed number density fluctuation field of lens galaxies is
given by

$\displaystyle\delta_{\rm g}(\chi_{\rm l},\chi_{\rm l}\bm{\theta})$
$\displaystyle=\delta_{\rm g}^{\rm int}(\chi_{\rm l},\chi_{\rm l}\bm{\theta})$
$\displaystyle\hskip 20.00003pt+2(\alpha_{\rm mag,l}-1)\kappa(\chi_{\rm
l},\chi_{\rm l}\bm{\theta})$ (14)

where $\delta_{\rm g}^{\rm int}$ is the intrinsic number density fluctuation
field, $\alpha_{\rm mag}$ is the power-law slope of galaxy counts around a
given magnitude cut (see main text), and $\kappa(\chi_{\rm
l}\bm{\theta},\chi_{\rm l})$ is the convergence field that is the projected
mass density field up to $z_{\rm l}$ in the direction $\bm{\theta}$, given by

$\displaystyle\kappa(\chi_{\rm l},\chi_{\rm l}\bm{\theta})=\int_{0}^{\chi_{\rm
l}}{\rm d}\chi W(\chi,\chi_{\rm l})\delta(\chi,\chi_{\rm l}\bm{\theta})$ (15)

with the lensing efficiency function

$\displaystyle W(\chi,\chi_{\rm l})\equiv\frac{3}{2}H_{0}^{2}\Omega_{\rm
m}a^{-1}\chi\frac{\chi_{\rm l}-\chi}{\chi_{\rm l}}.$ (16)

The same large-scale structure, characterized by $\kappa$, also distorts
images of source galaxies used in the galaxy-galaxy weak lensing measurements,
and therefore the magnification bias adds additional statistical scatter to
the measurements.

Extending the method in Oguri and Takada (2011) and Shirasaki and Takada
(2018), we derive the additional contribution due to magnification bias to the
covariance matrix between the lensing profiles for the lens samples at $z_{\rm
l}$ and $z_{\rm l^{\prime}}$, i.e. $\Delta\\!\Sigma(R_{n};z_{\rm l})$ and
$\Delta\\!\Sigma(R_{n^{\prime}};z_{\rm l^{\prime}})$:

$\displaystyle\delta{\rm Cov}[\Delta\\!\Sigma(R_{n},z_{\rm
l}),\Delta\\!\Sigma(R_{n^{\prime}},z_{\rm l^{\prime}})]$ $\displaystyle=$
$\displaystyle\frac{1}{\Omega_{\rm
s}}\int\\!\frac{\ell\mathrm{d}\ell}{2\pi}~{}\hat{J}_{2}\\!\left(\ell\frac{R_{n}}{\chi_{\rm
l}}\right)\hat{J}_{2}\\!\left(\ell\frac{R_{n^{\prime}}}{\chi_{\rm
l^{\prime}}}\right)\Sigma_{\rm cr}(z_{\rm s},z_{\rm l})\Sigma_{\rm cr}(z_{\rm
s},z_{\rm l^{\prime}})$ $\displaystyle\times\left\\{\left(2(\alpha_{\rm
l^{\prime}}-1)C_{{\rm g}\kappa_{\rm l}}(\ell;z_{\rm l},z_{\rm
l^{\prime}})+2(\alpha_{\rm l}-1)C_{{\rm g}\kappa_{\rm l}}(\ell;z_{\rm
l^{\prime}},z_{\rm l})\right)\left(C_{\kappa_{\rm s}\kappa_{\rm
s}}(\ell)+\frac{\sigma_{\epsilon}^{2}}{\bar{n}_{\rm s}}\right)\right.$
$\displaystyle\hskip 20.00003pt+2(\alpha_{l^{\prime}}-1)C_{{\rm g}\kappa_{\rm
s}}(\ell;z_{\rm l})C_{\kappa_{\rm l}\kappa_{\rm s}}(\ell;z_{\rm
l^{\prime}})+2(\alpha_{\rm l}-1)C_{\kappa_{\rm l}\kappa_{\rm l}}(\ell;z_{\rm
l})C_{{\rm g}\kappa_{s}}(\ell;z_{\rm l^{\prime}})$ $\displaystyle\left.\hskip
20.00003pt+4(\alpha_{\rm l}-1)(\alpha_{\rm l^{\prime}}-1)\left[C_{\kappa_{\rm
l}\kappa_{\rm l}}(\ell;z_{\rm l},z_{\rm l^{\prime}})\left(C_{\kappa_{\rm
s}\kappa_{\rm s}}(\ell)+\frac{\sigma_{\epsilon}^{2}}{\bar{n}_{\rm
s}}\right)+C_{\kappa_{\rm l}\kappa_{\rm s}}(\ell;z_{\rm l})C_{\kappa_{\rm
l}\kappa_{\rm s}}(\ell;z_{\rm l^{\prime}})\right]\right\\}.$ (17)

In this derivation, we assumed that the lens galaxies are at a single
redshift, $z_{\rm l}$ or $z_{\rm l^{\prime}}$, for simplicity, and we consider
the case with $z_{\rm l^{\prime}}\geq z_{\rm l}$ without loss of generality.
The auto- and cross-angular power spectra for the lensing convergence field
and the projected field of lens galaxies in the above equation are defined as

$\displaystyle C_{{\rm g}\kappa_{\rm s}}(\ell;z_{\rm l})=\frac{W(\chi_{\rm
l},\chi_{\rm s})}{\chi_{\rm l}^{2}}P_{\rm gm}\left(\frac{\ell}{\chi_{\rm
l}},z_{\rm l}\right)$ (18) $\displaystyle C_{{\rm g}\kappa_{\rm
l}}(\ell;z_{\rm l},z_{\rm l^{\prime}})=\frac{W(\chi_{\rm l},\chi_{\rm
l^{\prime}})}{\chi_{\rm l}^{2}}P_{\rm gm}\left(\frac{\ell}{\chi_{\rm
l}},z_{\rm l}\right)\Theta(z_{\rm l^{\prime}}-z_{\rm l})$ (19) $\displaystyle
C_{\kappa_{\rm l}\kappa_{\rm s}}(\ell,z_{\rm l})=\int{\rm
d}\chi\frac{W(\chi,\chi_{\rm s})W(\chi,\chi_{\rm l})}{\chi^{2}}P_{\rm mm}^{\rm
NL}\left(\frac{\ell}{\chi},z\right)$ (20) $\displaystyle C_{\kappa_{\rm
s}\kappa_{\rm s}}(\ell)=\int{\rm d}\chi\frac{W(\chi,\chi_{\rm
s})^{2}}{\chi^{2}}P_{\rm mm}^{\rm NL}\left(\frac{\ell}{\chi},z_{\rm l}\right)$
(21) $\displaystyle C_{\kappa_{\rm l}\kappa_{\rm l}}(\ell,z_{\rm l},z_{\rm
l^{\prime}})=\int{\rm d}\chi\frac{W(\chi,\chi_{\rm l})W(\chi,\chi_{\rm
l^{\prime}})}{\chi^{2}}P_{\rm gm}^{\rm NL}\left(\frac{\ell}{\chi},z\right),$
(22)

where $\Theta(x)$ is the Heaviside step function; $\Theta(x)=1$ if $x>0$,
otherwise $\Theta(x)=0$, and $\hat{J}_{2}(x)$ is the 2nd-order Bessel function
averaged within the separation bin, defined for the $n$-th separation bin
($R_{n}$) as

$\displaystyle\hat{J}_{2}(kR_{n})=\frac{2}{R_{\rm n,min}^{2}-R_{\rm
n,max}^{2}}\int_{R_{\rm n,min}}^{R_{\rm n,max}}R{\rm d}R~{}J_{2}(kR).$ (23)

with $k=\ell/\chi$ in Eq. (17). Note that we employed Limber’s approximation
(Limber, 1953) for the above angular power spectra. We also note that the
cross-correlation given by $C_{{\rm g}\kappa_{\rm l^{\prime}}}(\ell;z_{\rm
l},z_{\rm l^{\prime}})$ (Eq. 19) arises because the convergence field, which
causes the density modulations for the lens sample at $z_{\rm l^{\prime}}$
($\geq z_{\rm l}$), includes the contribution of the mass distribution at
$z_{\rm l}$, leading to a cross-correlation between the mass distribution and
the lens galaxy distribution that are both at $z_{\rm l}$. The additional
contribution due to the lensing magnification turns out not to be significant
compared to the total power of the covariance matrix, but we included it in
our cosmology analysis for completeness.

##  Appendix B Convergence tests of nested sampling

We use MultiNest for sampling in our parameter estimation. There are
uncertainties in the parameter estimation due to the sampling process itself,
because the size of the sampled chain is finite and the sampling itself
depends on the seed of MultiNest. In order to make sure that the uncertainty
due to MultiNest, $\Delta^{\rm MN}$, is small compared to the statistical
uncertainty, $\sigma^{\rm stat.}$, we run eight independent chains with
different MultiNest seeds. We confirm that the ratio of the sampler
uncertainty to the statistical error is $\Delta^{\rm MN}_{S_{8}}/\sigma^{\rm
stat.}_{S_{8}}=8.3\times 10^{-5}$; thus this source of uncertainty is
negligible.

We also check that our choice of the MultiNest hyper-parameters are suitable
for sampling the target posterior distribution by using the public code,
nestcheck (Higson et al., 2019). The result is shown in Fig. 15, which shows
the results of two independent chains. We can see that both chains are well
converged to the peak of the $S_{8}$ distribution, there is no significant
difference between these chains in the $(\log X,S_{8})$ plane, and the
uncertainty of the marginalized posterior distribution of $S_{8}$ estimated
from bootstrap resamples of either chain is not large. Thus we conclude that
the chains have fully converged with our choice of MultiNest hyper-parameters.

Figure 15: The result of nestcheck (Higson et al., 2019). Two independent
chains with different seeds indicated with different colors are used. The
upper right panel shows the posterior mass as a function of logarithmic prior
volume corresponding to the iteration number of MultiNest, where the prior
volume, $\log X$, decreases as the iteration number increases. The contour
shows the uncertainty of the posterior mass of the chain estimated from the
bootstrap from each chain. The lower left panel shows the posterior
distribution of the $S_{8}$ value, with uncertainty contour estimated from
bootstrap resamples of the chain. The lower right panel shows the 2-d
distribution of $\log X$ and $S_{8}$.

##  Appendix C Posterior distributions in a full parameter space

Fig. 16 shows the 1-d and 2-d posterior distributions in a full parameter
space in our baseline analysis, including the derived parameters, $\Omega_{\rm
m},\sigma_{8}$ and $S_{8}$.

Figure 16: The 1-d and 2-d posterior distributions over the full parameter
space for the baseline analysis shown in Fig. 4.

##  Appendix D Supplementary information for the results of different analysis
setups

In this appendix, we show supplemental systematic tests to explore some of the
systematics we found in Section VI.1.

###  D.1 Flat peak in $P(\Omega_{\rm m})$ of baseline analysis

For our baseline results shown in Fig. 4, the 1-d marginalized posterior of
$\Omega_{\rm m}$ exhibits a flat-shaped peak. We find that this is caused by
cosmic variance between the three lens samples. If we ignore the correlation
between the samples, the posterior for the combined samples is equivalent to
the product of the posteriors from each sample alone. This is illustrated in
Fig. 17, which shows the posterior (20 and 40% credible regions) in the
$\Omega_{\rm m}-\omega_{\rm c}$ subspace for the CMASS1-alone analysis and the
analysis without the CMASS1 sample. The contours of the two do not overlap.
The joint posterior has peaks at low and high values of $\Omega_{\rm m}$,
which explains why we find a flat-shaped peak when we marginalize over
$\omega_{\rm c}$.

Figure 17: Marginalized posterior distributions (20 and 40% credible regions)
of the “CMASS1 alone” analysis (blue) and “w/o CMASS1” analysis (green) in the
$(\Omega_{\rm m},\omega_{\rm c})$ plane. The red contour is the product of the
two marginalized posteriors, which shows two peaks in $\Omega_{\rm m}$.

###  D.2 A test for systematic effects in Ephor AB

As shown in Fig. 7, when we used the Ephor AB photo-$z$ catalog we found
significantly larger values of $\sigma_{8}$ and $S_{8}$ than in our baseline
analysis. We found that the number of source-lens pairs for the CMASS2 sample
in the Ephor AB catalog is $\sim 35\%$ smaller than that in the fiducial
photo-$z$ catalog (MLZ), and the lensing signal measured in Ephor AB catalog
has larger statistical errors. In particular, Fig. 18 shows that the galaxy-
galaxy lensing signal for the CMASS2 sample, measured from the Ephor AB
catalog, displays larger amplitudes at $R>12[h^{-1}{\rm Mpc}]$ than do the
other catalogs, which explains why we find larger values of $\sigma_{8}$ and
$S_{8}$. For a further check we confirm that the constraints on $\sigma_{8}$
and $S_{8}$ for the Ephor AB catalog become consistent with the results of our
baseline analysis if we do not use the lensing signal of the CMASS2 sample in
the parameter inference, as shown in Fig. 19.

Figure 18: The difference between the galaxy-galaxy weak lensing signals of
the CMASS2 sample using different photo-z catalogs. Because the number of
source-lens pairs in the Ephor AB catalog is $\sim 35$% smaller than that of
the fiducial (MLZ) catalog, the signal in Ephor AB is noisy, and happens to be
a several-sigma deviation from that in the fiducial catalog. Figure 19:
Comparison of cosmological parameter constraints from the signal measured in
the Ephor AB catalog to that of the fiducial catalog. When the noisy CMASS2
sample in the Ephor AB catalog is removed from the data vector, the
constraints on $\sigma_{8}$ and $S_{8}$ are consistent with the fiducial
result.