Cosmological inference from emulator based halo model I: Validation tests with HSCand SDSS mock catalogs

Hironao Miyatake,1, 2, 3, ∗ Yosuke Kobayashi,3, 4 Masahiro Takada,3, † Takahiro Nishimichi,5, 3 Masato Shirasaki,6, 7

Sunao Sugiyama,3, 4 Ryuichi Takahashi,8 Ken Osato,9 Surhud More,10, 3 and Youngsoo Park3

1Institute for Advanced Research, Nagoya University, Nagoya 464-8601, Japan2Division of Particle and Astrophysical Science, Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan

3Kavli 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, Japan4Physics Department, The University of Tokyo, Bunkyo, Tokyo 113-0031, Japan

5Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan6National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan7The Institute of Statistical Mathematics, Tachikawa, Tokyo 190-8562, Japan

8Faculty of Science and Technology, Hirosaki University,3 Bunkyo-cho, Hirosaki, Aomori 036-8561, Japan

9Institut d’Astrophysique de Paris, Sorbonne Universite, CNRS,UMR 7095, 98bis boulevard Arago, 75014 Paris, France

10The Inter-University Centre for Astronomy and Astrophysics, Post bag 4, Ganeshkhind, Pune 411007, India(Dated: January 5, 2021)

We present validation tests of emulator-based halo model method for cosmological parameterinference, assuming hypothetical measurements of the projected correlation function of galaxies,wp(R), and the galaxy-galaxy weak lensing, ∆Σ(R), from the spectroscopic SDSS galaxies and theHyper Suprime-Cam Year1 (HSC-Y1) galaxies. To do this, we use Dark Emulator developedin Nishimichi et al. based on an ensemble of N -body simulations, which is an emulation pack-age enabling a fast, accurate computation of halo clustering quantities (halo mass function, haloauto-correlation and halo-matter cross-correlation) for flat-geometry cold dark matter cosmologies.Adopting the halo occupation distribution, the emulator allows us to obtain model predictions of ∆Σand wp for the SDSS-like galaxies at a few CPU seconds for an input set of parameters. We presentperformance and validation of the method by carrying out Markov Chain Monte Carlo analyses ofthe mock signals measured from a variety of mock catalogs that mimic the SDSS and HSC-Y1 galax-ies. We show that the halo model method can recover the underlying true cosmological parametersto within the 68% credible interval, except for the mocks including the assembly bias effect (althoughwe consider the unrealistically-large amplitude of assembly bias effect). Even for the assembly biasmock, we demonstrate that the cosmological parameters can be recovered if the analysis is restrictedto scales R & 10 h−1Mpc (i.e., if the information on the average mass of halos hosting SDSS galaxiesinherent in the 1-halo term of ∆Σ is not included). We also show that, by using a single populationof source galaxies to infer the relative strengths of ∆Σ for multiple lens samples at different redshifts,the joint probes method allows for self-calibration of photometric redshift errors and multiplicativeshear bias. Thus we conclude that the emulator-based halo model method can be safely appliedto the HSC-Y1 dataset, achieving a precision of σ(S8) ' 0.04 after marginalization over nuisanceparameters such as the halo-galaxy connection parameters and the photo-z error parameter, andour method is complementary to methods based on perturbation theory.

I. INTRODUCTION

Wide-area imaging galaxy surveys offer exciting op-portunities to address the fundamental questions in cos-mology such as the nature of dark matter and the originof cosmic acceleration [1]. The current-generation imag-ing surveys such as the Subaru Hyper Suprime-Cam [2][HSC 3–5], the Dark Energy Survey [6] [DES 7], and theKilo-Degree Survey [KiDS 8] have succeeded in using ac-curate measurements of weak gravitational lensing effectsto obtain tight constraints on cosmological parameters.

∗ hironao.miyatake@iar.nagoya-u.ac.jp† masahiro.takada@ipmu.jp

Interestingly, the cosmological model inferred from theselarge-scale structure probes show the so-called σ8- or S8-tension [e.g. 4, 9] compared with cosmological modelsinferred from the Planck cosmic microwave background(CMB) measurement [10], perhaps indicating a signaturebeyond the standard cosmological model, i.e. the flat-geometry ΛCDM model [e.g. 9]. Upcoming galaxy sur-veys such as the Subaru Prime Focus Spectrograph [11][12], the Dark Energy Spectrograph Instrument [13], theVRO Legacy Survey of Space and Time [14], the ESAEuclid [15] and the NASA Roman Space Telescope [16][17] are expected to deliver a decisive conclusion on thepossible tension and also revolutionize our understandingof the Universe.

Main challenges of large-scale structure probes lie inuncertainty in galaxy bias, which refers to unknown re-

arX

iv:2

101.

0011

3v1

[as

tro-

ph.C

O]

31

Dec

202

0

2

lation between the distributions of matter and galaxiesin large-scale structure [18, 19]. Since physical processesinherent in formation and evolution of galaxies are stilldifficult to accurately model from the first principles, weneed both observational and theoretical approaches totackle the galaxy bias uncertainty in order for us to ob-tain “unbiased” and “precise” estimation of the under-lying cosmological parameters from large-scale structureobservables.

For observational approach, joint probes cosmology of-fers a promising way to mitigate the impact of galaxybias uncertainty on cosmology inference [7, 8, 20–26].In particular, galaxy-galaxy weak lensing, obtained bycross-correlating positions of foreground (lens) galaxieswith shapes of background galaxies, can be used to in-fer the average mass distribution around lens galaxies.A combination of galaxy-galaxy weak lensing with auto-correlation function of the same sample of galaxies as thelens galaxies can be used to observationally disentanglethe galaxy bias and the correlation function of the under-lying matter distribution from the measured clusteringsignal of galaxies.

On theory side, there are mainly two empirical ap-proaches to tackle the galaxy bias uncertainty. Firstone is a model based on perturbation theory (PT) oflarge-scale structure [27, 28]. As long as only the large-scale information of clustering observables in the linear orquasi-nonlinear regime is used and nuisance parametersto model galaxy bias are introduced, such a PT-basedmethod is expected to serve as an “accurate” theoreti-cal template of galaxy clustering [26, 29–31]. Accuracyhere refers to the fact that a PT model can reproducethe observed clustering correlation of galaxies by varyingthe bias parameters, down to a certain scale still in thequasi-nonlinear regime, where PT is valid. An advantageof this method is that the model can be used for any typeof galaxies, because PT is formulated based on propertiesof gravity and primordial fluctuations [32] and the freebias parameters absorb large-scale clustering propertiesof galaxies irrespective of galaxy types. A price to pay isthat the method breaks down at scales below a certainnonlinear scale, and cannot be used to extract cosmolog-ical information from the small-scale clustering signals,which generally carry higher signal-to-noise ratios thanin the large scales.

An alternative theoretical method is the halo modelapproach [33–36]. Halos are places where galaxies likelyform, and clustering properties of halos are relativelywell understood, on both analytical approach and N -body simulations [37]. Then an empirical model such asthe halo occupation distribution (HOD) method [38, 39]can be used to connect halos to galaxies. An advan-tage of this method is that it would allow one to use thesmall-scale information in cosmology inference, therebyyielding tighter constraints on cosmological parameters.However, a danger is that, if the model is not sufficientlyaccurate nor flexible enough to capture the complicatedgalaxy-scale physics, the method might lead to a signifi-

cant bias in cosmological parameters, more than the sta-tistical credible interval. A worst-case scenario is thatone might claim a wrong cosmology, e.g. a time-varyingdark energy model, from a given dataset due to the in-accurate theoretical templates.

Hence, the purpose of this paper is to assess perfor-mance and limitation of the halo model method for cos-mology inference. To do this, we use the Dark Emula-tor developed in Nishimichi et al. [40], which enables afast, accurate computation of halo clustering quantities(halo mass function, halo auto-correlation function andhalo-matter cross-correlation) for an input set of cosmo-logical parameters within flat-geometry wCDM frame-work with adiabatic Gaussian initial conditions. TheDark Emulator is particularly useful for our halo modelapproach, as it enables accurate predictions for the clus-tering quantities well into the non-linear regime andthereby allows us to make robust use of the cosmolog-ical information from small scales. We combine DarkEmulator with the HOD model to make model pre-dictions of the projected correlation functions of galax-ies, wp(R) and the galaxy-galaxy weak lensing, ∆Σ(R),that mimic those measured from the spectroscopic SDSSDR11 galaxies [41] and the HSC Year1 (HSC-Y1) galaxies[42]. More precisely, we consider mock galaxies of LOWZand CMASS galaxies in the redshift range 0.15 . z . 0.7for the spectroscopic galaxies in the wp measurement andfor the lens sample in ∆Σ, and then consider mock galax-ies of the deep HSC-Y1 data for the background galaxysample in ∆Σ. We use a variety of mock catalogs for theSDSS galaxies to assess performance and limitation of thehalo model method for cosmology inference, including amock where we implement an extreme version of galaxyassembly bias [43]. Here we quantify the performance bystudying whether the halo model method can recover thecosmological parameters employed in the mock catalogs,after marginalization over the halo-galaxy connection pa-rameters – hereafter we will often refer to this exerciseas cosmology challenges. Moreover, following Oguri andTakada [21], we assess the ability of our method to per-form self-calibration of photometric redshift errors andmultiplicative bias error for the joint probes cosmology.Those errors are among the most important, systematicerrors for weak lensing cosmology, and we show belowthat the method allows for an accurate self-calibrationof the errors, i.e. a robust estimation of cosmologicalparameters mitigating the impact of the systematic er-rors. This paper gives a validation of the halo modelmethodology for cosmology inference that we are plan-ning to carry out with the actual SDSS and HSC datasets(Miyatake et al. in preparation). The results of this pa-per are also compared to those of the companion paper,Sugiyama et al. [26], which used a perturbation theory in-spired method and was validated against the same mockcatalogs.

The structure of this paper is as follows. In Section IIwe briefly review the halo model for wp and ∆Σ and dis-cuss the methodology of how cosmological parameters

3

can be estimated from the clustering observables. InSection III we describe details of N -body simulations, thehalo catalogs, the mock catalogs of SDSS and HSC galax-ies, and Dark Emulator. In Section IV we describeour strategy for assessing performance and limitation ofthe halo model method, i.e. cosmology challenges donefrom the comparison of the halo model with the mocksignals. In Section V, we show the main results of thispaper. Section VI is devoted to conclusion and discus-sion. Throughout this paper, unless otherwise stated, weemploy the flat-geometry Planck cosmology as a targetcosmology for cosmology challenges and as a fiducial cos-mology when we compute cosmological dependences ofobservables. The model is characterized by Ωm = 0.3156(the present-day matter density parameter), h = 0.672and σ8 = 0.831, respectively, and we adopt the units ofc = 1 for the speed of light.

II. THEORY

In this paper we focus on two observables that are ob-tained from imaging and spectroscopic data of galaxiesin the overlapping regions of the sky. One is the galaxy-galaxy weak lensing that can be measured by stackingshapes of background galaxies around a sample of fore-ground lensing galaxies. Here we assume that the back-ground galaxy sample is taken from the Subaru HSC im-ages, while the foreground lensing galaxies are from theSDSS spectroscopic galaxies. The other is the projectedcorrelation function of the spectroscopic galaxies that arefrom the same population of galaxies used as lens (fore-ground) galaxies in the galaxy-galaxy weak lensing.

A. Observables: galaxy-galaxy weak lensing andprojected auto-correlation function

Galaxy-galaxy weak lensing [24, 44] probes the aver-aged excess surface mass density profile, ∆Σ, around thelensing (foreground) galaxies that is given in terms of thesurface mass density profile, Σgm(R), as

∆Σ(R; zl) = 〈Σgm〉(< R)− Σgm(R)

= Σcrit(zl, zs) γ+(R)|R=χl∆θ, (1)

where γ+ is the average tangential shear of backgroundgalaxies in the circular annulus of projected centric ra-dius R from the foreground galaxies, and χl is the comov-ing angular diameter distance to each foreground galaxy.Note that background galaxy shapes are averaged overall the pairs of foreground-background galaxies in thesame projected separation R, not the angular separation∆θ, even when the lensing galaxies have a redshift dis-tribution. 〈Σgm〉(< R) is the average surface mass den-sity within a circular aperture of radius R, defined as

〈Σgm〉(< R) ≡ 1/(πR2)∫ R

02πR′dR′ Σgm(R′). Σcrit is

the critical surface mass density that describes a lensing

efficiency for pairs of foreground/background galaxies asa function of their redshifts, and is defined as

Σcrit(zl, zs) ≡χs(zs)

4πGχls(zl, zs)χl(zl)(1 + zl), (2)

where χs and χls are the comoving angular diameter dis-tances to each source galaxy and between source and lensgalaxies in each pair, respectively. The factor (1 + zl) isdue to our use of the comoving coordinates. In practicewe need to take into account the redshift distributionsof lens and source galaxies that are straightforward toinclude, e.g. following the method in Ref. [24].

The surface mass density profile around lensing galax-ies, Σgm(R), is given in terms of the three-dimensionalgalaxy-matter cross-correlation function as

Σgm(R; zl) = ρm0

∫ ∞−∞

dπ ξgm

(√π2 +R2; zl

)(3)

where ρm0 is the mean mass density today, ξgm(r) is thethree-dimensional cross-correlation function between thegalaxies and matter, and π is the separation along theline-of-sight direction, and R is the projected separationperpendicular to the line-of-sight direction. Here we ig-nored the contribution from the background mean massdensity because it is not relevant for weak lensing observ-ables [compared to Eq. 3 in Ref. 26]. Fourier-transformedcounterpart of ξgm(r) is the galaxy-matter cross powerspectrum, Pgm(k), defined as

ξgm(r; zl) ≡∫ ∞

0

k2dk

2π2Pgm(k; zl)j0(kr), (4)

where j0(x) is the zero-th order spherical Bessel function.Hereafter we omit the redshift of the lensing galaxies, zl,in the argument of the correlation functions for nota-tional simplicity. Throughout this paper we assume adistant observer approximation for computations of pro-jected correlation functions. In other words, we ignorethe effects of curved sky for definitions of separationsalong or perpendicular to the line-of-sight direction. Theexcess surface mass density profile is given in terms ofthe galaxy-matter cross power spectrum as

∆Σgm(R) ≡ 〈Σgm〉(< R)− Σgm(R)

= ρm0

∫ ∞0

k2dk

2π2Pgm(k)J2(kR), (5)

where J2(x) is the 2nd-order Bessel function.To obtain the theoretical template of ∆Σ(R) for a given

cosmological model within the flat ΛCDM framework, weuse Dark Emulator, developed in Ref. [40], that en-ables accurate and fast computations of halo statisticalquantities; the halo mass function, the halo-matter cross-correlation function and the halo-auto correlation func-tion as a function of redshift, halo masses, and separa-tions for an input cosmological model. As demonstrated

4

in Ref. [40], the emulator outputs the predictions to bet-ter than a few percent in the fractional amplitude overscales of separations we are interested in.

However, the emulator does not take into account theeffects of baryonic physics, and this is a limitation weshould keep in mind. Nevertheless we do not think thislimitation causes a catastrophic failure of our approachdue to the following reasons. The baryonic physics is lo-cal in the sense that it affects the matter distribution atscales smaller than a maximum scale, denoted as R∗. Forexample, even if we consider a violent effect of the AGNfeedbacks, it would affect the mass distribution aroundhalos up to a few Mpc at maximum. As nicely discussedin Refs. [45, 46], we can safely consider that the bary-onic effect causes a “redistribution” of matter at r . R∗around galaxies, and does not alter the mass distributionat r & R∗ (r is the three-dimensional radius from the cen-ter of galaxy). In other words, even in the presence ofthe baryonic physics, the mass conservation at r . R∗holds. Keeping this in mind, we can say that Dark Em-ulator, as designed, can accurately model the matterdistribution at r & R∗ around the host halos of galaxies,and also satisfies the mass conservation to within r ' R∗,even though Dark Emulator ceases to accurately pre-dict the mass profile at r . R∗.

Based on the above consideration, we can rewrite thelensing profile (Eq. 5) as

∆Σ(R) = 〈Σgm〉(< R)− Σgm(R)

=M(< R∗)

πR2+

2

R2

∫ R

R∗

R′dR′ Σgm(R′)− Σgm(R).

(6)

In the second line on the r.h.s., we used the relation

M(< R∗) ≡∫ R∗

02πR′dR′ Σgm(R′), where M(< R∗) is

the mass interior of a circular aperture of radius R∗. Thesecond term is the contribution over the range of [R∗, R]to the average mass density. As long as we focus on thelensing profiles at R ≥ R∗, Dark Emulator can accu-rately predict the lensing profile, including the interiormass at the aperture of R ' R∗. On the other hand, weneed to introduce various nuisance parameters to modelthe lensing “profile” at R ≤ R∗, if the information is in-cluded, and then marginalize over the parameters whenestimating the interior mass M(< R∗) and performingcosmology inference.

Another clustering observable we use is the projectedauto-correlation function for spectroscopic galaxy sam-ple that is the same sample used as lens (foreground)galaxies in the galaxy-galaxy weak lensing measurement.The projected correlation function is defined by a line-of-sight projection of the three-dimensional auto-correlationfunction of galaxies, ξgg(r), as

wp(R) ≡ 2

∫ πmax

0

dΠ ξgg

(√R2 + Π2

), (7)

where πmax is the length of the line-of-sight projection.Note that the projected correlation function has the unit

of [h−1Mpc]. Throughout this paper, unless explicitlystated, we employ πmax = 100 h−1Mpc as our defaultchoice. Also notice that ξgg(r) is given in terms of theauto-power spectrum of the galaxy number density field,Pgg, as

ξgg(r) =

∫ ∞0

k2dk/(2π2) Pgg(k)j0(kr). (8)

The projected correlation function is not sensitive to theredshift-space distortion (RSD) effect due to peculiar ve-locities of galaxies, if a sufficiently large projection length(πmax) is taken [see Fig. 6 in Ref. 47]. The RSD effectitself is a useful cosmological probe, but its use requiresan accurate modeling [43, 48], which is not straightfor-ward. Hence, the projected correlation function makes itsomewhat easier to compare with theory in a cosmolog-ical analysis. In the following, we ignore the RSD effectin most cases of our cosmology challenges, but will sepa-rately discuss the impact of the RSD effect in parameterestimation.

B. Theoretical template: Dark Emulatorimplementation of halo model

In this section we describe details of the halo modelimplementation to make model predictions for the ob-servables ∆Σ(R) and wp(R). As can be found aroundEqs. (5) and (7), we need the real-space cross-power spec-trum of galaxies and matter Pgm(k; z), and the real-spaceauto-power spectrum of galaxies, Pgg(k; z), as a functionof cosmological models to compute the observables.

1. Halo Occupation Distribution

In the halo model we assume that all matter is asso-ciated with halos, and the correlation function of matteris given by the contributions from pairs of matter in thesame halo and those in two different halos, which arereferred to as the 1- and 2-halo terms, respectively. Toconnect halos to galaxies in a given sample, we employthe halo occupation distribution [HOD 34, 36, 38]. TheHOD model gives the mean number of central and satel-lite galaxies in halos of mass M as

〈N〉(M) = 〈Nc〉(M) + 〈Ns〉(M), (9)

where 〈 〉(M) denotes the average of a quantity for halosof mass M .

We employ the mean HOD for central galaxies, givenas

〈Nc〉(M) =1

2

[1 + erf

(logM − logMmin

σlogM

)], (10)

where erf(x) is the error function and Mmin and σlogM

are model parameters.

5

For the mean HOD of satellite galaxies, we employ thefollowing form:

〈Ns〉(M) ≡ 〈Nc〉(M)λs(M) = 〈Nc〉(M)

(M − κMmin

M1

)α,

(11)

where κ,M1 and α are model parameters, and we have in-troduced the notation λs(M) = [(M−κMmin)/M1]α. Forour default prescription we assume that satellite galaxiesreside only in a halo that already hosts a central galaxy.This means Nc = 1 for such halos. Then we assumethat the number of satellite galaxies in halos of massM follows the Poisson distribution with mean λs(M);Prob(Ns) = (λs)

Ns exp(−λs)/Ns! for halos that host acentral galaxy inside, and Prob(Ns) = δKNs,0

for halos that

do not host a central galaxy, where δKij is the Kroneckerdelta function. Under these assumptions, the mean num-ber of galaxy pairs living in the same halo with mass M ,which is relevant for the 1-halo term, can be computedas

〈N(N − 1)〉 = 〈Nc〉 〈N(N − 1)〉|Nc=1

+ (1− 〈Nc〉) 〈N(N − 1)〉|Nc=0

= 〈Nc〉 [〈(Nc +Ns)(Nc +Ns − 1)〉]Nc=1

= 〈Nc〉[⟨N2

s

⟩+ 〈Ns〉

]= 〈Nc〉

[2λs + λ2

s

]. (12)

where we have used the fact N = 0 for halos which haveno central galaxy for our default prescription.

We have 5 model parameters, Mmin, σlogM , κ,M1, α,to characterize the central and satellite HODs in total foreach galaxy sample for a given cosmological model.

Once the HOD model is given, the mean number den-sity of galaxies in a sample is given as

ng =

∫dM

dnh

dM[〈Nc〉(M) + 〈Ns〉(M)] , (13)

where dnh/dM is the halo mass function which gives themean number density of halos in the mass range [M,M+dM ]. In the following we use “blue” fonts to denote aquantity that can be supplied by Dark Emulator. Inthe above case we use Dark Emulator to compute thehalo mass function for wCDM cosmology, while we inputthe HOD model, as described, to compute the galaxynumber density.

2. Galaxy-galaxy weak lensing profile

As we described above, the galaxy-galaxy weak lensingarises from the cross-correlation of (spectroscopic) lens-ing galaxies with the surrounding matter distribution inlarge-scale structure, ξgm(r; zl). Hence we need to modelξgm(r) as a function of the parameters of halo-galaxyconnection and cosmological parameters.

100

101

102

R∆

Σ[1

06 M

pc−

1 ]

total

central

central w/offset

satellite

100 101 102

R [h−1Mpc]

100

101

102

103

104

Rw

p[h−

2M

pc2 ]

total

1-halo cen-sat

1-halo sat-sat

2-halo cen-cen

2-halo cen-sat

2-halo sat-sat

FIG. 1. Break down of different contributions to the galaxyweak lensing profile (∆Σ(R); upper panel) and the projectedcorrelation function of galaxies (wp; lower) for the SDSSLOWZ-like galaxies at z = 0.251, which are computed us-ing Dark Emulator and our model ingredients of the halo-galaxy connection for the Planck cosmology. For illustrativepurpose we multiply each observable by R so that their dy-namic range (y-axis) becomes narrower. Upper panel: Thedashed line shows a contribution arising from the cross-correlation of central galaxies with the surrounding matterdistribution, while the dot-dashed line denotes a contributionfrom the cross-correlation of satellite galaxies with matter.For comparison, we also show how a possible off-centeringof central galaxies affects the lensing profile, although we donot include this effect in the theoretical templates. Here, asa working example, we consider the off-centering parameterspoff = 0.3 and Roff = 0.4 irrespective of halo mass, where poff

models a fraction of off-centered central galaxies in halos ofmass M , while R is the off-centering radius relative to thescale radius of NFW profile of halos. The upper solid linesis the total power. Lower: The similar break down of differ-ent contributions to the projected correlation function of thegalaxies, for the same model in the upper panel.

Since the Fourier space gives a somewhat simpler formof the expression, we mainly write down the power spec-trum that is the Fourier-transformed counterpart of thetwo-point correlation function, here ξgm. Under the halomodel approach, the cross-power spectrum of galaxiesand matter is given as

Pgm(k) =1

ng

∫dM

dnh

dM[〈Nc〉(M) + 〈Ns〉(M)us(k;M, z)]

× Phm(k), (14)

6

where Phm(k) is the halo-matter cross-power spectrumthat can be computed by Dark Emulator as a func-tion of cosmological model within wCDM cosmologies.The quantity us(k;M) is the Fourier transform of theaveraged radial profile of satellite galaxies in host ha-los of M at redshift z, which we need to specify. Notethat all the quantities in the above equation are evalu-ated at the lens redshift zl, but we omit to denote inthe argument of each function for notational simplicity.Dark Emulator was built to calibrate the halo-mattercross-correlation by measuring the averaged mass profilearound halos with mass M in N -body simulation outputs[40]. By construction, the halo-matter cross-correlationsatisfies the mass conservation around halos. We alsonote that Dark Emulator already includes both the1- and 2-halo term contributions in Phm, which corre-spond to the cross-correlations of halos with matter inthe same halo and the surrounding matter outside thehalo, respectively.

More exactly speaking, Dark Emulator outputsξhm(r;M) for an input set of parameters (halo mass, sep-aration and cosmological parameters) that is the Fouriertransform of Phm(k;M). In order to obtain ξgm or ∆Σfor the assumed model we extensively use the publicly-available FFTLog[49] code to perform the Hankel trans-forms when going back and forth between real- andFourier-space (see Eqs. 4 and 5).

For the radial profile of satellite galaxies, we as-sume that satellite galaxies follow a Navarro-Frenk-White(NFW) profile [50] that is an approximated model of theaveraged dark matter profile in halos. To compute theNFW profile as a function of halo mass, redshift and cos-mological model, we need to specify the scaling relationof mass concentration with halo mass in the halo mat-ter profile. For this we employ the fitting formula givenin Ref. [51]; more exactly, we use the publicly-availablepython code, “Colossus”[52] [53], to compute the con-centration parameter for a given set of parameters (halomass, redshift and cosmological parameters).

As for the default model of the halo-galaxy connection,we consider neither the off-centering effect nor the incom-pleteness effect of galaxies, where the latter describes apossibility that some fraction of even very massive ha-los might not host a SDSS-like galaxy due to an incom-plete selection after the specific color and magnitude cuts[25, for details of the model]. Rather than introduc-ing additional nuisance parameters to model these ef-fects, we employ a minimum halo model as our baselinemodel. However, we use different types of mock cata-logs of SDSS-like galaxies including the off-centering ef-

fects and the incompleteness effect, and then will use themock catalogs to validate and assess the performance ofthe baseline method. If our baseline model can recoverthe underlying cosmological parameters, we claim thatthe method is validated. If the method shows any failureto recover the cosmological parameters, we will start tointroduce more parameters. However, if the cosmologicalconstraints turn to be sensitive to such details of a treat-ment of galaxies inside the halo, this is a sign of the failureor limitation of the halo model based method, becausesuch a small-scale distribution of galaxies is very diffi-cult to accurately model due to complexities of physicalprocesses inherent in formation and evolution of galax-ies. Hence, rather than including such model parameters,we employ the minimum halo model to assess its perfor-mance.

The upper panel of Fig. 1 shows how different terms ofPgm contributes to the ∆Σ profile for the SDSS-like galax-ies at z = 0.251 as we will describe in more detail. Eventhough satellite galaxies tend to reside in massive halos,all the curves have a similar shape (R-dependence) inlarge R bins, R & 10 h−1Mpc, for a fixed cosmology. Allthe small-scale physics involved in the halo-galaxy con-nection affects ∆Σ at R . 10 h−1Mpc. Note that, dueto the non-local nature of ∆Σ, the small-scale physics af-fects ∆Σ up to relatively large scales, compared to a virialradius of massive halos. Also importantly, the integratedlensing signal up to a few Mpc scales gives an estimate ofthe average halo mass, which has a close tie to the large-scale amplitudes of ∆Σ and wp via the scaling relation ofhalo bias with halo mass (see below). For comparison,the figure also shows the impact of off-centering effectsof central galaxies on ∆Σ, assuming that some fraction ofcentral galaxies might be off-set from the true halo cen-ter as a result of the assembly history of galaxies in theirhost halos [54, 55].

3. Projected correlation function of galaxies

To model wp we need to model the real-space corre-lation function of galaxies, ξgg, or the Fourier transformPgg for an input set of parameters. The auto-power spec-trum of galaxies in a sample is decomposed into the twocontributions, the 1- and 2-halo terms, and those aregiven within the halo model framework as

Pgg(k) ≡ P 1hgg (k) + P 2h

gg (k) (15)

with

7

P 1hgg (k) =

1

n2g

∫dM

dnh

dM〈Nc〉(M)

[2λs(M)us(k;M) + λs(M)2us(k;M)2

],

P 2hgg (k) =

1

n2g

[∫dM

dnh

dM〈Nc〉(M) 1 + λs(M)us(k;M)

]×[∫

dM ′dnh

dM ′〈Nc〉(M ′) 1 + λs(M

′)us(k;M ′)]Phh(k;M,M ′), (16)

where Phh(k;M,M ′) is the power spectrum between twohalo samples with masses M and M ′. Dark Emula-tor outputs the real-space correlation function of halos,ξhh(r;M,M ′), that is the Fourier transform of Phh. Simi-larly to the case of ∆Σ, we use the FFTLog code to performthe Fourier transform to obtain the prediction of ξgg(r)(see Eq. 8) and then perform the line-of-sight integral toobtain the prediction of wp(R) (see Eq. 7). We note that,strictly speaking, our standard halo model implementa-tion of the 1-halo term behaves as a shot noise like termof k0(= const.) at the limit of k → 0, and this gives asubtle violation of the mass and momentum conservation[32]. One could modify the halo model at very small k toenforce the conservation laws, as done in Ref. [56], but wefound that our treatment, practically, gives a sufficientlyaccurate model prediction over scales of separations thatwe are interested in, for the SDSS-like galaxies [48] [alsosee 57, for the similar discussion]. As given by Eq. (7),we employ πmax = 100 h−1Mpc, as our default choice tointegrate ξgg(r =

√R2 + π2) to obtain the model predic-

tion for wp(R).

The lower panel of Fig. 1 shows different contributionsin the projected correlation function wp for the Planckcosmology, assuming the same model as in the upperpanel. As can be found from Eq. (16), the 1-halo termarises from the central-satellite and satellite-satellite cor-relations, while the 2-halo term is from the correlationsof central-central, central-satellite and satellite-satellitegalaxies in different halos, respectively. The figure clearlyshows that all the contributions to the 1-halo term areconfined to small scales, at R . a few Mpc, a virial ra-dius of massive halos, reflecting the local nature of wp

in contrast to ∆Σ. All the different contributions to the2-halo term have a similar shape (R-dependence); thedifferent terms differ from each other only by a mul-tiplicative factor for a fixed cosmology. This meansthat the shape of the 2-halo term is not sensitive to de-tails of the halo-galaxy connection. The features aroundR ' 90 h−1Mpc are the BAO features, which appear atsmaller scales than the BAO scale of 100 h−1Mpc in thethree-dimensional correlation function, due to the line-of-sight projection. However, we note that we will notinclude the BAO information in the parameter estima-tion in the following; we will use the wp information upto R ' 30 h−1Mpc as our default choice.

C. Joint probes cosmology: mitigation of galaxybias uncertainty

For convenience of our discussion let us define thecross-correlation coefficient function for halo correlationfunction:

rhm(r) ≡ ξhm(r)

[ξhh(r)ξmm(r)]1/2. (17)

As shown in Fig. 31 of Ref. [40], Dark Emulator pre-dicts rhm ' 1 on r & 10 h−1Mpc, the scale greaterthan a typical size of massive halos, and rhm is closeto unity within 5% or so even at the intermediate scalesa few Mpc < r . 10 h−1Mpc. As stressed in our com-panion paper [26], the real-space observables have an ad-vantage that all variants due to the halo-galaxy connec-tion are confined to small scales, r . 10 h−1Mpc, and thecross-correlation function on large scales satisfies rhm ' 1[also see 58, for the similar discussion]. This is not nec-essarily true in Fourier space, because the variations onsmall scales become extended in Fourier space due to thenature of Fourier transform. The simplest example is theshot noise; the shot noise affects the real-space correla-tions at zero separation, while it behaves like a whitenoise and affects the power spectrum over all scales (kbins).

It is also constructive to discuss an asymptotic be-havior of the halo correlation functions, ξgm and ξgg, onlarge scales. At scales much greater than the nonlinearscale, r R∗, the galaxy-matter cross correlation has anasymptotic behavior given as

ξgm(r) −−−−→rR∗

1

ng

∫dM

dnh

dM〈Ng〉(M)ξhm(r;M)

' beffξmm(r), (18)

where we have used us(k;M) → 1 for k 1/R∗, andwe defined the effective bias parameter, at a sufficientlylarge separation satisfying r R∗, as

beff ≡1

ngξmm(r)

∫dM

dnh

dM〈Ng〉 ξhm(r;M). (19)

In addition, as can be found from Eq. (3), ∆Σ, which hasa dimension of [hMMpc−2], has an additional depen-dence on ρm0

∆Σ ∝ ρm0ξgm ∝ Ωmbeffξmm. (20)

8

0.85

0.90

0.95

1.00

1.05

1.10

1.15

∆Σ/∆

Σfid

100 101

R [h−1Mpc]

0.85

0.90

0.95

1.00

1.05

1.10

1.15

wp/w

fid p

Ωm

σ28

5%

10%

FIG. 2. Dependences of ∆Σ and wp on cosmological parame-ters, σ8 (blue lines) and Ωm (red), computed using the DarkEmulator based halo model. Here we consider the HODparameters for the LOWZ sample at z = 0.251. Here weconsider fractional changes of σ2

8 (not σ8) or Ωm by ±5 % or±10 %, respectively, where the other parameters are fixed totheir fiducial values. The solid and dashed, respective linesshow the fractional changes in ∆Σ or wp relative to that forthe fiducial model when σ2

8 or Ωm is changed to a positive ornegative side from its fiducial value. The error bars aroundunity denote 1σ statistical errors that are computed from thediagonal terms of the covariance matrix expected for the SDSSand HSC-Y1 data, although the neighboring bins are highlycorrelated with each other.

Note that the dependence on h is not relevant becauseall the quantities are measured in units where the depen-dence of h is factorized out.

Similarly the auto-correlation function of galaxies isfound to have an asymptotic behavior at the limit of largescales, r R∗, as

ξgg(r) −−−−→rR∗

1

n2g

[∫dM

dnh

dM〈Ng〉(M)

]×[∫

dM ′dnh

dM ′〈Ng〉(M ′)

]ξhh(r;M,M ′)

' b2effξmm(r). (21)

This is not an exact relation, but we found that therelation rgg ≡ ξgm/[ξggξmm]1/2 ' 1 holds for scales ofr R∗, as shown in Fig. 32 of Ref. [40]. Thus the com-

bination of ξgm and ξgg, which are inferred from ∆Σ andwp, respectively, can be used to infer the underlying mat-ter correlation function ξmm, and in turn we can use itto extract cosmological information. Furthermore, as im-plied from Eq. (6), an amplitude of ∆Σ at scales arounda transition scale between the 1- and 2-halo terms givesan estimate of the average mass of halos hosting galaxies:

∆Σgm(R)|R'a few Mpc → Mh (22)

In turn this can put a strong constraint on the halo bias,beff , via the scaling relation of halo bias with halo mass(or the dependence of ξhm and ξhh amplitudes on halomass). Hence, combining the small- and large-scale infor-mation of ∆Σ with wp helps break degeneracies betweencosmological parameters and the galaxy bias, and thengives useful constraints on cosmological parameters. Thisis the basic picture of how the joint probes cosmologyusing ∆Σ and wp can constrain cosmological parameters.However, if the scaling relation of the large-scale galaxybias amplitude with the average halo mass is broken, thismethod does not work. This would be the case for theassembly bias effect, where the galaxy bias depends onsecondary parameter(s) related to the assembly historyof host halos, in addition to halo mass.

In Fig. 2, we study how ∆Σ and wp depend on thecosmological parameters, Ωm and σ8, because large-scalestructure probes are most sensitive to these parameters.Here we consider the expected signals of ∆Σ and wp forthe SDSS LOWZ-like galaxies at z = 0.251 (as we willdescribe in detail in Section III B) which are computedusing Dark Emulator based on the halo model. Wenote that, due to the non-local nature of ∆Σ, the lensingprofile at a particular scale, say R0, is sensitive to thematter-galaxy cross-correlation at r ∼ R0/2. Note thatwe employ Rcut = 2 and 3 h−1Mpc for the scale cuts of∆Σ and wp as our default choice, respectively, and willuse the information of ∆Σ and wp at scales greater thanthe scale cuts for the cosmology challenges.

Let us first consider the dependence of σ8. There aretwo competing effects in the change of our target observ-ables originating from that of σ8 for models with a fixedΩm. First, an increase of σ8 boosts the amplitude ofξmm by definition. Second, it leads to a decrease of halobias (beff) for massive halos hosting SDSS-like galaxies(∼ 1013h−1M), because a model with higher σ8 leadsto more evolved large-scale structure at an observed red-shift, thus the abundance of massive halos gets increasedat the redshift, and such halos become less biased trac-ers, resulting in a lowered bias amplitude compared tothat for the fiducial model. For ∆Σ, the first effect ismore significant; an increase of σ8 leads to an increase ofthe amplitude of ∆Σ over all the scales we consider. Forwp, these competing effects almost cancel out on largescales, R & a few Mpc in the 2-halo term, and the in-crease of σ8 does not largely change the amplitude onthe large scales. Nevertheless we should emphasize thatthe change of σ8 causes a scale-dependent modificationin these observables, especially wp, in contrast to the lin-

9

0.7

0.8

0.9

1.0

1.1

1.2

1.3

∆Σ/∆

Σfid

100 101

R [h−1Mpc]

0.7

0.8

0.9

1.0

1.1

1.2

1.3

wp/w

fid p

logMmin

σ2

20% 40%

0.7

0.8

0.9

1.0

1.1

1.2

1.3

∆Σ/∆

Σfid

100 101

R [h−1Mpc]

0.7

0.8

0.9

1.0

1.1

1.2

1.3

wp/w

fid p

logM1

α

κ

20%

40%

FIG. 3. Dependence of ∆Σ and wp on changes to HOD parameters can be seen in the top and the bottom panels, respectively.Here the fiducial HOD parameters are chosen to resemble the SDSS LOWZ-like galaxy sample at z = 0.251 for the Planckcosmology. When one HOD parameter is changed, the other parameters are fixed to their fiducial values. Solid line shows theresult for an increase (positive-side change) of each HOD parameter from its fiducial value, while dashed line is the result forthe decrease. From light- to dark-color lines show the results for the fractional change of each HOD parameter by ±20 % and±40 %, respectively. The left panel shows the dependences of the central HOD parameters, Mmin and σlogM , while the rightpanel shows the results for the satellite HOD parameters, M1, κ and α. The error bars are the same in the previous figure.

ear theory that predicts a constant (scale-independent)shift in the ratio at these large scales (also see Figure 6in [26] for the similar discussion). We checked that thescale-dependent changes are from a combination of theeffects of σ8 changes on matter clustering (ξgg) and thehalo bias function (see Fig. 19 in Appendix A). Theseeffects are automatically built in the Dark Emultorpredictions. Also note that the positive- and negative-side changes of σ8 cause an asymmetric change in wp atlarge scales. On the other hand, the 1-halo term ampli-tude of wp is boosted because the increase of σ8 leads tothe increased abundance of massive halos, more satellitegalaxies reside in such halos, and their correlations add.Thus the change of σ8 leads to characteristic modifica-tions of the amplitude and scale dependence in ∆Σ andwp. The notable features around R ∼ 90 h−1Mpc aredue to the effect on BAO features [59], but this is notrelevant for our results because we use the informationof ∆Σ and wp on scales up to 30 h−1Mpc for the fidu-cial choice, so do not include the BAO information inparameter estimation.

Next we discuss the results for the change of Ωm, fora fixed σ8. First, an increase of Ωm leads to a fasterevolution of clustering growth for models with a fixed σ8

(the fixed normalization today), so leads to the smalleramplitude of ξmm at this redshift (z = 0.251). However,recalling that ∆Σ ∝ Ωm 〈γ+〉 ∼ Ωmδm, an increase of Ωm

leads to higher amplitudes in ∆Σ due to the prefactor,while it leads to smaller amplitudes in wp. In addition,the change of Ωm causes a scale-dependent modificationin both ∆Σ and wp.

In Fig. 3 we study how a change in each of the HODparameters alters ∆Σ and wp for the Planck cosmology.Here we employ the same HOD parameters for the LOWZgalaxies as in Fig. 2, and vary only one HOD parameterfor each result, where other parameters are fixed to theirfiducial values. The figure shows that, when each of thecentral HOD parameters is varied, it causes a significantchange over all the scales including both the 1- and 2-halo terms, because the change in the parameter leadsto a change in the mean halo mass. More precisely, anincrease in Mmin or a decrease in σlogM leads to an in-crease in the mean halo mass, leading to the increasedamplitudes in ∆Σ and wp. The fractional changes in ∆Σand wp at large separations in the 2-halo term regimeare roughly given as δwp/wp ' 2δ∆Σ/∆Σ, reflecting thefacts ξgg ' b2effξmm and ξgm ' beffξmm at the large scalelimit as we discussed above. In addition, it is clear that

10

the 1-halo term amplitude of ∆Σ is sensitive to the cen-tral HOD parameters, physically to the mean mass ofhalos hosting the galaxies in a sample. Hence, combiningthe 2-halo term amplitudes of ∆Σ and wp allows one tobreak degeneracies between the bias parameter and otherparameters, and then adding the 1-halo term informa-tion of ∆Σ can constrain the HOD parameters and thentighten the determination of the large-scale bias. Thesatellite HOD parameters change the relative contribu-tion of massive halos to ∆Σ and wp as satellite galaxiespreferentially reside in massive halos. In particular, theparameter M1, which determines the amplitude of thesatellite HOD, causes a decent change in the large-scaleamplitudes of ∆Σ and wp. The effects of other parame-ters are relatively mild.

Comparing Figs. 2 and 3 manifests that the cosmo-logical parameters and the HOD parameters lead to dif-ferent changes in ∆Σ and wp. Hence combining theseobservables in the 1- and 2-halo term regimes allows foran efficient determination of the cosmological parame-ters, by breaking the parameter degeneracies. These fig-ures nicely illustrate the complementarity of ∆Σ and wp,which is the main focus of this paper.

D. Observational effects: geometrical cosmologydependence, photo-z, multiplicative shear bias, and

RSD

In this section we discuss the three observational ef-fects, i.e. the geometrical dependence of the observableson cosmology, photometric redshift errors of source galax-ies, multiplicative shear bias and redshift-space distortioneffect. When comparing the measured signals with themodel templates, we need to include these effects in themodel templates, and here we describe how to do this.

1. Geometrical cosmology dependence

The observables we consider in this paper are ∆Σ(R)and wp(R), which are different from other possiblechoices of the lensing and clustering observables such asγ+(θ) and wp(θ). Measurements of ∆Σ and wp requirean observer to assume a reference cosmology to performa correction of the lensing efficiency Σcr(zl, zs) (see Eq. 1)as well as the conversion of angular scales (θ) and redshiftdifferences to the projected separation (R) and radial sep-aration (π). However, the assumed cosmology generallydiffers from the true underlying cosmology, and this de-pendence needs to be taken into account. We use themethod in Ref. [60] to include this effect.

For a flat-geometry ΛCDM model, the relevant param-eter is only Ωm (or Ωde that is the density parameter ofdark energy), because it affects the angular and radialdistances (the lensing efficiency also depends on the com-bination of angular diameter distances and the overallfactor of Ωm). Note that the dependence on h is taken

out from the observables because all the quantities aremeasured in units of h−1Mpc or hM Mpc−2 for wp and∆Σ, respectively. The Ωm dependences of ∆Σ and wp

through this effect turn out to be very small, but add aslight sensitivity of Ωm in the theoretical templates.

2. Photometric redshift errors

Photometric redshift errors of source galaxies are oneof the most serious systematic errors in the weak lens-ing measurements. An accuracy of photometric redshifts(hereafter often simply photo-z), delivered from a set ofbroad-band filters (grizy in the HSC data), is limited,and can never be perfect, compared to spectroscopic red-shifts, although the photo-z accuracies are calibrated us-ing the COSMOS catalog [61]. For lens redshift we as-sume spectroscopic redshifts as we will focus on the spec-troscopic SDSS-like galaxies. Here we discuss how we cantreat a possible uncertainty of source redshifts in the ob-servable ∆Σ. In this paper we use the method proposedin Oguri and Takada [21]. In this method we use a “sin-gle” population of source galaxies, selected based on thephoto-z, and then use the relative strengths of ∆Σ formultiple lens samples at different redshifts to calibratethe photo-z uncertainty of source galaxies in a statisticalsense.

In the presence of a redshift distribution of sourcegalaxies, an estimator of ∆Σ(R) from the measured el-lipticity component of each source galaxy is given, e.g.by Eq. (11) in Ref. [24], as

∆Σ(R) =1

2R

∑l,s wls

[⟨Σ−1

cr

⟩ls

]−1es+∑

l,s wls(23)

where R is the “responsivity” that is needed to con-vert the measured galaxy ellipticity, defined in terms of(a2−b2)/(a2 +b2) (a, b is the major, minor axes when thegalaxy shape is approximated by an ellipse), to the lens-ing shear, defined in terms of (a− b)/(a+ b) [23, 42, 62];es+ is the tangential ellipticity component of the s-thsource galaxy with respect to the l-th lensing galaxy; wls

is the weight (see below). The summation∑

l,s runs overall pairs of source and lens galaxies that are in a givenbin of projected separation, R = χ(zl)∆θ. The measuredlensing signal is for the effective redshift of lens galaxies,zl.

In a case that we have only the posterior distributionof photometric redshift for each source galaxy, the “ef-fective” lensing efficiency needs to be estimated as

⟨Σ−1

cr

⟩ls≡∫ ∞

0

dzs Σ−1cr (zl, zs)p(zs), (24)

where p(zs) is the posterior distribution of redshift for thes-th source galaxy that satisfies the normalization condi-tion

∫∞0

dzs p(zs) = 1. Note that we set Σ−1cr (zl, zs) = 0

when zs < zl. We employ the weight wls, motivated by

11

the inverse-variance weight, to have a higher signal-to-noise ratio for the weak lensing measurement [63–65]:

wls =

[⟨Σ−1

cr

⟩(l,s)

]2e2

rms + σ2e

(25)

where erms is the rms intrinsic ellipticity per componentand σe is the measurement error of ellipticity for the s-thsource galaxy.

Thus, in an estimator of ∆Σ, the quantities⟨Σ−1

cr

⟩ls

and wls depend on photometric redshifts of source galax-ies via p(zs) for individual source galaxies. If the esti-mated redshift distribution of source galaxies is system-atically offset from the underlying true distribution, themeasured ∆Σ has a systematic bias (offset) from the trueone (even if the assumed cosmological model for the ∆Σestimation is the true cosmology). Following the methodin Refs. [21, 66], we take into account the systematic off-set by introducing additional nuisance parameter, ∆zph,to shift the posterior distribution of source redshift forall source galaxies as

p(zs) −→ p(zs + ∆zph). (26)

Then we can repeat the computations of⟨Σ−1

cr

⟩ls

andwls for the pairs of source and lens galaxies in actualSDSS and HSC-Y1 datasets. We found that the lensingprofile after shifting the source redshift distribution iswell approximated by the following multiplicative formas

∆Σ(R; zl,∆zph) ' fph(R; zl,∆zph)∆Σ(R; zl,∆zph = 0),(27)

where fph(R; zl,∆zph) is the multiplicative factor tomodel the effect of systematic photo-z error, and zl is themean of lens redshifts. Here we stress, by notation “”,that the above correction is made for the measured ∆Σ.By using a single population of source galaxies, we noticethat the shift ∆zph leads to changes in the amplitudes of∆Σ for each of the multiple lens samples (LOWZ andthe two subsamples of CMASS, divided into two redshiftbins) depending on the lens redshift (zl). Conversely, wecan use the relative variations in the ∆Σ amplitudes atdifferent lens redshifts to calibrate out ∆zph, simultane-ously with cosmological parameter estimation. This is aself-calibration method of photo-z errors as proposed inOguri and Takada [21]. In the above equation, we explic-itly include the R-dependence in the calibration factor,which could arise because the shift could change relativecontributions of different lens-source pairs to the lens-ing efficiency

⟨Σ−1

cr

⟩ls

. However, the R-dependence, al-beit very weak, appears only for the highest-redshift lenssample (CMASS2 in our sample, as we will define later).

For our method, we multiply the inverse of the calibra-tion factor in Eq. (27) with the theoretical template of∆Σ, rather than correcting for the measurement, for anassumed ∆zph:

∆Σ(R)→ ∆Σ(R)/fph(R; ∆zph). (28)

We then include ∆zph as additional nuisance parameterwhen carrying out the parameter inference. This is bet-ter because we will use the same covariance matrix inour parameter inference, which allows for apple-to-applecomparison of the performance for different setups (be-cause this method does not change the error bars of ∆Σin each R bin).

3. Multiplicative shear errors

An accurate weak lensing measurement requires anexquisite, accurate characterization of individual galaxyshapes. This is not straightforward [42], and an imper-fect shape measurement leaves a residual systematic er-ror in the weak lensing measurements. Systematic errorsin shape measurements are usually modeled by “multi-plicative” and “additive” biases in the measured galaxyellipticities, given as γ → (1+m)γ+c, where m and c arethe multiplicative and additive bias parameters, respec-tively [67]. Since the spatial positions of lensing galaxieson the sky are considered random with respect to the po-sitions of source galaxies, the galaxy-galaxy weak lensingis not affected by the additive bias [42]. Hence, in thepresence of the multiplicative shear bias, we modify thetheoretical template as

∆Σ(R) −→ (1 +m)∆Σ(R;m = 0). (29)

By working on the single sample of source galaxies, wecan employ the single m parameter for all the lensing pro-files at multiple lens redshifts. This is another advantageof the method of Ref. [21]. This is a good approxima-tion as long as source galaxies are well separated fromlens redshifts; this is the case as long as we can ignore acontamination of source galaxies into lens redshifts dueto photo-z errors. To make this method work, in actualdata, we will employ a conservative cut of source galaxyredshifts to ensure that source galaxies are well separatedfrom lens redshifts (Miyatake et al., in prep.).

4. Redshift-space distortion

To model the projected correlation function of galaxies,we ignored the redshift-space distortion (RSD) effect thatis caused by peculiar velocities of galaxies. For our de-fault choice of the projected length, πmax = 100 h−1Mpc,however, the RSD effect is not negligible. We followthe method proposed in Ref. [47]. We employ the lin-ear Kaiser formula [68] to model the RSD effect. Wefollow Eqs. (51)-(54) in Ref. [47] to model the redshift-space two-point correlation function of galaxies using theKaiser RSD β factor, given by β ≡ (1/beff)d lnD/d ln a,where beff is the effective linear bias for a sample of galax-ies (see Eq. 19) and D is the linear growth factor. Thenwe modify the theoretical template of wp as

wp(R)→ fRSD(R, πmax)wp(R;β = 0), (30)

12

TABLE I. The value of each cosmological parameter forthe fiducial Planck cosmology which we use in the follow-ing cosmology challenges (we will study whether the methodcan recover the true values within the credible interval).The column, labeled as “supporting range”, gives the sup-porting range of each parameter in Dark Emulator thatoutputs the halo clustering quantities for a flat-geometrywCDM model specified by a set of 6 cosmological parame-ters Ωde, ln

(1010As

), ωb, ωc, ns, w each of which should be

in the supporting range. Here σ8 and S8 ≡ σ8(Ωm/0.3)0.5 arederived parameters, and the values in table are those for thefiducial Planck cosmology.

Parameters fiducial value supporting range [min,max]Ωde 0.6844 [0.54752, 0.82128]ln(1010As

)3.094 [2.4752, 3.7128]

ωb 0.02225 [0.0211375, 0.0233625]ωc 0.1198 [0.10782, 0.13178]ns 0.9645 [0.916275, 1.012725]w −1 [−1.2,−0.8]σ8 0.831 derivedS8 0.852 derived

where fRSD(R, πmax) is the correction multiplicative fac-tor to account for the RSD effect. This factor dependson the projected separation R and the projection lengthπmax for an assumed cosmology. For our default choiceof πmax = 100 h−1Mpc, the linear Kaiser factor is a goodapproximation to model the RSD effect on wp [also see26].

III. N-BODY SIMULATIONS, DARKEMULATOR, AND MOCK CATALOGS OF HSC

AND SDSS-LIKE GALAXIES

In this paper we use two types of mock catalogsof SDSS- and HSC-like galaxies. First, we use high-resolution N -body simulations, with periodic boundaryconditions, and the halo catalogs to generate the mockcatalogs of SDSS galaxies. We then measure ∆Σ and wp

from the mock catalogs to define the mock signals thatwe use in cosmology challenges. Second, we also use themock catalogs of SDSS- and HSC-like galaxies built inthe light-cone simulations, including the simulated lens-ing signals on the HSC-like source galaxies, and use thosecatalogs to estimate the covariance matrix of the observ-ables. These mocks are the same as those used in ourcompanion paper, Sugiyama et al. [26]. In this section,we describe details of N -body simulations and the mockcatalogs.

A. N-body simulations and Dark Emulator

In this paper we extensively use Dark Emulator de-veloped in Nishimichi et al. [40], which is a software pack-age enabling fast, accurate computations of halo clus-

tering quantities for a given cosmological model. Herewe will briefly review Dark Emulator. They con-structed an ensemble of cosmological of N -body simu-lations, each of which was performed with 20483 parti-cles for a box with 1 or 2 Gpc/h on a side length, for101 cosmological models within the flat wCDM cosmolo-gies. The wCDM cosmology is parametrized by 6 cosmo-logical parameters, p = ωb, ωc,Ωde, ln

(1010As

), ns, w,

where ωb(≡ Ωbh2) and ωc(≡ Ωch

2) are the physical den-sity parameters of baryon and CDM, respectively, h isthe Hubble parameter, Ωde ≡ 1 − (ωb + ωc + ων)/h2 isthe density parameter of dark energy for a flat-geometryuniverse, As and ns are the amplitude and tilt param-eters of the primordial curvature power spectrum nor-malized at kpivot = 0.05 Mpc−1, and w is the equa-tion of state parameter for dark energy, respectively. Forthe N -body simulations, they included the neutrino ef-fect fixing the neutrino density parameter ων ≡ Ωνh

2 to0.00064 corresponding to 0.06 eV for the total mass ofthree neutrino species. They included the effect of mas-sive neutrinos only in the initial linear power spectrum[see 40, for details]. To carry out “cosmology challenges”in the following, we employ the fiducial Planck cosmologythat is characterized by the parameter values in Table I.We use the N -body simulation realizations of 1 h−1Gpcbox size for the fiducial Planck cosmology to constructthe mock catalogs for SDSS-like galaxies. The mass ofsimulation particle for the fiducial Planck simulations ism = 1.02 × 1010 h−1M. In the following we use haloswith mass greater than 1012 h−1M, corresponding toabout 100 simulation particles.

For each N -body simulation realization (each redshiftoutput) for a given cosmological model, they constructeda catalog of halos using Rockstar [69] that identifieshalos and subhalos based on clustering of N -body par-ticles in phase space (position and velocity space). Thespherical overdensity mass, with respect to the halo cen-ter that is defined from the maximum mass density,M ≡ M200m = (4π/3)R3

200m × (200ρm0), is used for def-inition of halo mass, where R200m is the spherical haloboundary radius within which the mean mass density is200 times ρm0. By combining the outputs of N -bodysimulations and the halo catalogs at multiple redshifts inthe range z = [0, 1.48], they built an emulator, named asDark Emulator, which enables fast and accurate com-putations of the halo mass function, halo-matter cross-correlation, and halo auto-correlation as a function ofhalo masses, redshift, separations and cosmological mod-els. It was shown that Dark Emulator achieves a suf-ficient accuracy for these statistical quantities for halosof 1013M, which is a typical mass of host halos of SDSSgalaxies, compared to the statistical measurement errorsof ∆Σ and wp expected from the HSC and SDSS data,as shown in Fig. 31 of the paper. In summary, DarkEmulator outputs

• dnh

dM (M ; z,p): the halo mass function for halos inthe mass range [M,M + dM ] at redshift z

13

TABLE II. Specifications of the mock galaxy catalogs thatresemble the LOWZ and CMASS galaxies for the spectro-scopic SDSS DR11 data. For the CMASS sample we considertwo subsamples divided into two redshift ranges. We give theredshift range and the comoving volume for each sample, as-suming 8300 deg2 for the area coverage. The column denotedas “representative redshift” is a representative redshift of eachsample which we assume to represent the clustering observ-ables for each sample (i.e. ignore redshift dependences of theobservables within the redshift bin). The lower columns, be-low double lines, denote the HOD parameters used to buildthe mock catalogs of each sample from N -body simulationsfor the Planck cosmology. Note that Mmin and M1 are inunits of h−1M.

LOWZ CMASS1 CMASS2redshift range [0.15, 0.35] [0.47, 0.55] [0.55, 0.70]representative redshift 0.251 0.484 0.617volume [(h−1Gpc)3] 0.67 0.81 2.00

HOD parameters fiducial valueslogMmin 13.62 13.94 14.19σlogM 0.6915 0.7919 0.8860κ 0.51 0.60 0.066logM1 14.42 14.46 14.85α 0.9168 1.192 0.9826

• ξhm(r;M, z,p): the halo-matter cross-correlationfunction for a sample of halos in the mass range[M,M + dM ] at redshift z

• ξhh(r;M,M ′, z,p): the halo-halo auto-correlationfunction for two samples of halos with masses[M,M + dM ] and [M ′,M ′ + dM ′] at redshift z

for an input set of parameters, halo mass M (and M ′ forthe correlation function of two halo samples), redshift z,and cosmological parameters p. In addition, the DarkEmulator package outputs auxiliary quantities, basedon emulation, such as the linear halo bias (the large-scalelimit of the halo bias), the linear mass power spectrum,the linear rms mass fluctuations of halo mass scale M(σLm(M)), and σ8. The supporting range of each of cos-mological parameters for Dark Emulator is given inTable I. These ranges are sufficiently broad, e.g. to coverthe range of cosmological constraints from the currentstate-of-the-art large-scale structure probe such as theSubaru HSC cosmic shear results [4, 5]. In this paperwe will use Dark Emulator to perform MCMC analy-ses of cosmological parameters by comparing the modeltemplates of ∆Σ and wp with the mock signals expectedfrom the SDSS and HSC data.

B. Mock catalogs for the SDSS-like galaxies for themock signals of ∆Σ and wp

1. Mock catalogs of SDSS LOWZ- and CMASS-like galaxies

We use the N -body simulation realizations and thehalo catalogs for the Planck cosmology to build mock

catalogs of galaxies that resemble spectroscopic galaxiesin the SDSS-III BOSS DR11 sample [70]. We considerthree galaxy samples in three redshift bins: “LOWZ”galaxies in the redshift range z = [0.15, 0.30] and twosubsample of “CMASS” galaxies that are obtained fromsubdivision of CMASS galaxies into two redshift bins,z = [0.47, 0.55] and z = [0.55, 0.70]. Here we considerluminosity-limited samples rather than flux-limited sam-ples for these galaxies (see Miyatake et al. in prepara-tion for details) [also see 24, for the similar discussion onthe stellar-mass limited sample]. The luminosity-limitedsample is considered as a nearly volume-limited samplein each redshift bin, and we expect that properties ofgalaxies do not strongly evolve within the redshift bin,which is desired because we ignore the redshift evolu-tion of clustering observables within the redshift bin foreach sample. Table II summarizes characteristics of eachgalaxy sample.

To build the mock catalogs for each galaxy sample,we first perform a fitting of the HOD model predictionsto the projected correlation function wp that is actuallymeasured from the SDSS data for each sample, assumingthe Planck cosmology model, and then estimate the HODparameters for the best-fit model. Note, however, thatwe here use the analytical halo model method used inMore et al. [25], and did not use the weak lensing profileto further constrain the HOD parameters. The fiducialHOD model is the same as that described in Section II B.Table II gives the best-fit HOD parameters for each sam-ple.

We use the outputs of N -body simulations at z =0.251, 0.484 and 0.617 as representative redshifts of thegalaxy samples to build the mock catalogs. We thenpopulate galaxies into halos of each realization for thePlanck cosmology, using the best-fit HOD parameters inTable II [also see 43, 48, for details of the method]. Forour default mocks, we assume the NFW profile for theradial distribution of satellite galaxies. The default mockis the same as the default HOD model used in the the-oretical template (see Section II B), so we will use thedefault mock to perform a sanity check of whether wecan recover the true cosmological parameters, i.e. theparameter values of Planck cosmology, in the parameterestimation by comparing model templates with the mocksignals measured from the default mocks. In the defaultmocks, we neither include the off-centering effect of cen-tral galaxies, an incompleteness effect of central galaxies,nor the redshift-space distortion effect. For other mocks,we include these effects one by one, and then study theimpact of each effect on parameter estimation.

2. Mock signals of ∆Σ and wp

We generate the mock signals of ∆Σ and wp for theSDSS-like galaxies, by measuring those clustering observ-ables from each of the mock catalog realizations for thePlanck cosmology as we described in the preceding sec-

14

5

10

15

20

R∆

Σ[1

06 M

/pc]

LOWZ CMASS1 CMASS2

100 101 102

R [Mpc/h]

0

100

200

300

400

Rw

p[(

Mp

c/h

)2 ]

100 101 102

R [Mpc/h]100 101 102

R [Mpc/h]

FIG. 4. Mock signals of ∆Σ and wp for the SDSS LOWZ, CMASS1 and CMASS2 galaxies at the representative redshifts,z = 0.251, 0.484 and 0.617, respectively. The signals are computed from the mock catalogs of the galaxies that are constructedby applying the HOD model (Table II) to halo catalogs in high-resolution N -body simulations (see text for details). Theerror bars in each bin denote the statistical errors expected from the SDSS DR11 data (8,300 sq. deg.) and the HSC-Y1 data(140 sq. deg.), which are the square root of the diagonal components of each covariance matrix that is estimated from the mockcatalogs of HSC and SDSS galaxies in the light-cone simulations. Here we multiply R by ∆Σ(R) and wp(R) for illustrativepurpose.

tion. In doing this, we take an advantage of the periodicboundary conditions in each mock, which allows for a fastcomputation of the clustering quantities using the FFTalgorithm. In addition, the measured clustering quanti-ties are not affected by the window function thanks tothe periodic boundary conditions. We here describe de-tails of the measurement method of ∆Σ and wp from eachmock catalog [43, 48].

For ∆Σ, we first project the matter (N -body) parti-cles and galaxies along one axis of N -body cubic boxassuming that the axis is along the line-of-sight direc-tion, and assign the matter particles and galaxies to the46, 3322 two-dimensional grids using the Nearest GridPoint (NGP) interpolation kernel. We then Fouriertransform the matter and galaxy density fields and take a

product of the two, Re[δ2Dg (k⊥)δ2D

m (−k⊥)], where k⊥ is

the two-dimensional wavevector. We perform the inverse2D Fourier transform to obtain

ξ2Dgm(R) =

∫d2k⊥(2π)2

eik⊥·R Re[δ2Dg (k⊥)δ2D

m (−k⊥)],

(31)

where R is the two-dimensional separation vector per-pendicular to the projection direction. We perform theazimuthal angle average in an annulus of each radial binand then obtain the two-dimensional galaxy-matter cross

correlation function, ξ2Dgm(R).

We compute the projected surface mass density profilearound galaxies from ξ2D

gm(R), according to Eq. (3), as

Σgm(R) ' ρm0ξ2Dgm(R). (32)

The above surface mass density differs from Eq. (3) by aconstant additive term (spatially-homogeneous term), asthis is okay because such a constant term is irrelevant tothe weak lensing shear or the excess surface mass densityprofile. To achieve higher spatial resolution, we mea-sure Σgm(R) using the folding method of the FFT box[71]. Following the method described in Refs. [72, 73],we perform multiple measurements in which we fold thebox different times (we denote the folding times as nfold).We have six measurements of nfold = 0, 1, 2, 3, 4, 5 foreach catalog, while the grid number 46, 3322 is kept un-changed at each FFT step. It means that we have mea-surements which have finer resolutions by up to a factorof 25 = 32. We then combine the Σgm(R) signals ob-tained by stitching the six measurements between thefive boundary scales 0.125, 0.25, 0.5, 0.1, 0.2h−1 Mpc.Then we compute the excess surface mass density profile,∆Σ(R) from Σ(R), according to Eq. (1). In each mea-surement, to achieve better statistics we perform threemeasurements using the projection along x-, y- or z-axisdirections, and use the average of the three results as the

15

measured signal of ∆Σ for the realization.For wp, we first assign the mock galaxies to the 10243

three-dimensional FFT grids using the NGP kernel. Wethen Fourier transform the number density field, and takeits square amplitudes |δg(k)|2 for each wavevector k. Byperforming the inverse Fourier transform, we obtain theestimate of the correlation function at each spatial sepa-ration r = (R, π),

ξgg(r) =

∫d3k

(2π)3eik·r

∣∣∣δg(k)∣∣∣2 . (33)

Then we estimate the projected correlation function,wp(R), from the azimuthal angle average in the circularannulus of each radial bin R and the line-of-sight projec-tion over π = [0, πmax], according to Eq. (7).

We use 19 and 22 independent realizations that arebuilt using the different seeds of the initial conditionsfor ∆Σ and wp, respectively [40] [74]. We then measurethe average mock signals of ∆Σ and wp for each of thegalaxy samples (Table II). These correspond to 19 and22 (h−1Gpc)3 volumes, respectively, that are larger thanthe volume of any of the three redshift slices in Table IIby at least a factor of 11. Recalling that the overlappingarea of HSC-Y1 data and SDSS is only about 140 sq.deg., compared to the SDSS area coverage of 8,300 sq.deg., the effective volume for the ∆Σ measurement ofeach sample is smaller than that of wp by a factor of 60.Hence the simulation volume used for the mock signalof ∆Σ is larger than that of the ∆Σ measurement by atleast a factor of 650. Thus by using the mock signals forthe larger volumes in cosmology challenges, we can min-imize any unwanted bias in estimated parameters due tosample variance. In this way we can evaluate the per-formance of each method, i.e. the ability to recover thetrue cosmological parameters, without being affected bythe sample variance.

The data points in each panel of Fig. 4 show the mocksignals of ∆Σ and wp for each of the LOWZ, CMASS1and CMASS2 samples (see Table II) for the Planck cos-mology. The error bars in each bin are the statisticalerrors expected for the SDSS and HSC-Y1 surveys as wewill explain below. The figure shows that a sufficientnumber of the realizations of the mock catalogs lead towell-converged, smooth signals in each bin, and the sta-tistical scatters appear to be negligible. This allows usto robustly evaluate the performance of the method forcosmological parameter estimation.

C. Light-cone mock catalogs of HSC- andSDSS-like galaxies for estimating the covariance

As we described in Section II A, ∆Σ is independent ofsource redshift, and depends only on the galaxy-mattercross correlation at lens redshift. Based on this fact, weconstruct the best-available, accurate mock signal for ∆Σfrom the mock catalogs of SDSS-like galaxies as we de-scribed above. However, the lensing effects on the same

TABLE III. The cumulative signal-to-noise (S/N) ratios of∆Σ, wp and the joint measurements for the LOWZ, CMASS1and CMASS2 samples, which are estimated using the mocksignals and the covariance matrices. Here we define the “cu-mulative” S/N over the ranges of R/[h−1 Mpc] = [3, 30] and[2, 30] for ∆Σ and wp, respectively, which are our baselinechoices of the radial range (see text for details). For the “to-tal” S/N of ∆Σ we take into account the cross-covariancesbetween ∆Σ’s of different galaxy samples. We assume thatthe wp-signals for the three samples are independent fromeach other, and ignore the cross-covariances between ∆Σ andwp.

LOWZ CMASS1 CMASS2 total∆Σ 8.64 8.86 8.48 14.7wp 32.9 32.1 30.4 54.7

joint (∆Σ + wp) 34.0 33.3 31.6 56.6

population of source galaxies by foreground structures atdifferent redshifts, from the redshifts of SDSS galaxies,– cosmic shear causes statistical scatters in the observedgalaxy ellipticitiies. We need to properly take into ac-count these effects. In this subsection, we describe themock catalogs of HSC- and SDSS-like galaxies that arebuilt in the light-cone simulations, and then use the mockcatalogs to model the covariance matrices of ∆Σ and wp.

To construct the mock catalogs in a light-cone volume,we use the full-sky, light-cone simulations generated inTakahashi et al. [75]. The light-cone simulation consistsof multiple spherical shells with an observer being at thecenter of the sphere, and each spherical shell contains thelensing fields and the halo distribution, where the lensingfields at the representative redshift of the shell can beused to simulate the lensing distortion effect on a galaxyat the position by foreground structure if the galaxy islocated within the shell. The halo distribution in eachshell reflects a realization of halos in large-scale structureat the redshift corresponding to the radius of the shell(the distance from an observer to the shell). In this paperwe use 108 realizations of the full-sky simulations.

As described in Appendix B in detail, we populateHSC- and SDSS-like galaxies in the full-sky, light-conesimulation. For the HSC galaxies, we use the actualHSC shape catalog [42], used for the HSC-Y1 weak lens-ing measurements [4], and populate each galaxy into thecorresponding shell in the light-cone simulation accord-ing to its angular position (RA and dec) and photometricredshift (best-fit photo-z). Then we simulate the lensingsignal on each galaxy using the lensing information oflight-cone simulation. Thus the mock HSC catalog in-cludes properties of actual data (the angular positionsand the distributions of ellipticities and photo-z’s) aswell as the geometry and masks of the HSC footprints.Since the HSC-Y1 data still has a small area coverage(140 sq.deg.), we identify 21 footprints of the HSC-Y1data in each of the all-sky, light-cone simulation realiza-tions [76, 77]. We thus generate 2268 mock catalogs ofthe HSC data in total.

16

For the SDSS galaxies, we populate galaxies into halosin the light-cone simulation based on the HOD method.We built mock catalogs for each of the three SDSS-likegalaxies (LOWZ, CMASS1, and CMASS2) in their corre-sponding redshift ranges in the assigned survey regions ofthe SDSS DR11 survey footprints. Given the large sur-vey area of SDSS data (about 8,000 sq. deg.), we identifyonly one SDSS region in each realization of the light-conesimulations, and thus build 108 mock catalogs for eachof the SDSS galaxies in total.

Each of our mock catalog realizations in the light-conevolume not only contains the angular and redshift (ra-dial) distributions of HSC and SDSS galaxies, but alsocontains the lensing effects on each source galaxy by theSDSS galaxies and other foreground structures at dif-ferent redshifts from the SDSS redshift. We then per-form measurements of ∆Σ and wp from each realizationusing the same analysis pipelines that are used in theactual measurements of the real HSC and SDSS data.Finally we estimate the covariance for the galaxy-galaxyweak lensing ∆Σ for each of the SDSS LOWZ, CMASS1and CMASS2 galaxies, from the scatters among the 2268measurements. The covariance matrices of ∆Σ for thedifferent galaxy samples at different lens redshifts havecross-covariance components because the measurementsuse the same source galaxies and are affected by weaklensing due to the same foreground structure (cosmicshear) in each light-cone simulation realization. Our co-variance properly takes into account the cross-covariance.On the other hand, for the covariance of wp, we applythe jackknife method to estimate the covariance of wp

for each realization, and then use the average of the 108covariances as the estimate of the wp covariance used inthis paper.

The error bars in each bin in Fig. 4 are estimated fromthe covariance matrices we described above. The figureshows that the HSC-Y1 data allows for a significant de-tection of the lensing signals at each bin. Table III givesthe cumulative signal-to-noise (S/N) ratios expected formeasurements of ∆Σ and wp from the HSC-Y1 and SDSSdata and the joint measurements. Here, as for our de-fault choice of the range of separation scales, we adopt3 ≤ R/[h−1Mpc] ≤ 30 for ∆Σ and 2 ≤ R/[h−1Mpc] ≤ 30for wp, respectively. The table gives the cumulative S/Nintegrated over the separation ranges properly taking intoaccount the covariance and the cross-covariance matrices.It is clear that wp has a grater S/N value than ∆Σ doesby a factor of 4, because of the much wider area coverageof SDSS data compared to the HSC-Y1 data by a factorof 60. Nevertheless, we will show later that combining∆Σ and wp is crucial to lift parameter degeneracies andobtain useful cosmological constraints.

In the following, we do not include the cosmology de-pendence of the covariance matrix, motivated by the dis-cussion in [78]. Hence the differences in the performanceof different setups/methods that we will show below arepurely from the differences in the model or setups.

TABLE IV. Prior range of each model parameter. In cos-mology challenges of parameter estimation, we consider twocosmological parameters, Ωde, and ln

(1010As

), and consider

5 HOD parameters for each of the LOWZ, CMASS1 andCMASS2 samples. Hence we have 17 model parameters intotal for the baseline method. For an extended method, wealso include the nuisance parameters to model the effects ofphoto-z errors (∆zph) and the multiplicative shear bias (mγ)for which we employ the Gaussian prior with width given inthe number in the table.

Parameters prior range [min,max]Ωde [0.54752, 0.82128]ln(1010As

)[2.4752, 3.7128]

logMmin [12.0,14.5]σ2

logM [0.01,1.0]logM1 [12.0,16.0]κ [0.01,3.0]α [0.5,3.0]∆zph Gauss: 0.04 or 0.1mγ Gauss: 0.01

IV. METHODOLOGY FOR COSMOLOGYCHALLENGES

The purpose of this paper is to study whether the halomodel based method can recover the true cosmologicalparameters from a hypothetical parameter inference, i.e.comparing the theoretical templates with the mock sig-nals, taking into account the error covariance in the like-lihood analysis. Here we describe our methodology toperform cosmology challenges. We employ different se-tups of the analysis method to quantify the impact ofvarious effects on cosmological parameter estimation, anddescribe each setup and its purpose.

A. Parameter estimation method

We assume that the likelihood of the mock signalsgiven the model parameters follows a multivariate Gaus-sian distribution:

lnL(d|θ) = −1

2

∑i,j

[di − ti(θ)] (C−1)ij [dj − tj(θ)]

+ const., (34)

where d is the mock data vector, t is the model vectorcomputed from the theoretical template as a functionof the model parameters (θ), C−1 is the inverse of thecovariance matrix, and the summation runs over the in-dices, i, j, corresponding to the dimension of the datavector. In our baseline analysis we use, as the data vec-tor, ∆Σ(R) given in 9 radial bins, logarithmically-evenlyspaced over 3 ≤ R/[h−1Mpc] ≤ 30, and wp(R) in 16 ra-dial bins over 2 ≤ R/[h−1Mpc] ≤ 30, respectively, foreach galaxy sample [79]. Thus we use 75 data pointsin total (3 × (9 + 16) = 75). Note that, for our default

17

analysis, we do not include the abundance information ofgalaxies (ng) in parameter inference. Even if we employa weak prior on the abundance (e.g., 50% of the numberdensity), the following results remain almost unchanged,but will come back to a question of whether a mild useof the abundance information can improve the parameterestimation.

For the model parameters (θ) in Eq. (34), we considerthe two cosmological parameters, Ωde and ln

(1010As

),

and 5 HOD parameters for each of the LOWZ, CMASS1,and CMASS2 samples (Table II). Hence we have 17 pa-rameters (2 + 3 × 5 = 17) in total. Since the clusteringobservables are primarily sensitive to the amplitude pa-rameters Ωde and As for the flat ΛCDM cosmology, weconsider only the two parameters, and fix other cosmolog-ical parameters to their values of the Planck cosmologyin parameter inference. Hence the degrees of freedom forthe fiducial analysis is Ndof = Nd −Np = 75 − 17 = 58.If we include further nuisance parameters to model thephoto-z errors and/or the shear multiplicative bias, weinclude up to 19 parameters (i.e. 1 or 2 additional pa-rameters). For a given set of the model parameters, wecan compute the model vector t at each radial bin.

We then perform parameter estimation based on theBayesian inference:

P(θ|d) ∝ L(d|θ)Π(θ), (35)

where P(θ|d) is the posterior distribution of θ and Π(θ)is the prior distribution. Throughout this paper we em-ploy a flat prior on each model parameter as given inTable IV. We checked that the priors of the cosmologi-cal parameters are wide enough, and the following resultsfor the posterior distribution are not affected by the priorrange.

We draw samples from the posterior distribution of pa-rameters, given the mock signals, with the help of nestedsampling as implemented in the publicly available pack-age Multinest (Multi-Modal Nested Sampler) [80] to-gether with the package Monte Python [81] to samplethe posterior distribution of the parameters. In the fol-lowing we mainly focus on the posterior distributions ofΩm, σ8 and S8 ≡ σ8(Ωm/0.3)0.5. These are derived pa-rameters from the cosmological parameters we use (Ωde,ln(1010As

)), and we employ the definition of S8 follow-

ing Hikage et al. [4] so that the forecast of S8 estimationfrom our method is compared to the previous results.

In this paper we adopt the mode of the marginal-ized 1D or 2D posterior distribution to infer the centralvalue(s) of the parameter(s), and the highest density in-terval of the marginalized posterior to infer the credibleinterval of the parameter(s). We often report the best-fitparameters that correspond to a model at the maximumlikelihood in a multi-dimensional parameter space. Asstressed in [26] [also see 82], a point estimate of param-eter is not useful, because it is sensitive to the degree ofdegeneracies between parameters. For example, even ifwe consider an ideal case that the input signals are fromthe model predictions, the central value of a parameter,

estimated from the mode of the marginalized posteriordistribution, does not necessarily recover the true valueas a result of marginalization of the parameters, if thetarget parameter is highly degenerate with other param-eters. Rather a more useful quantity is the credible in-terval. Hence in the following we will mainly focus onthe credible interval, and evaluate each method/setup tostudy whether the true value of the cosmological param-eters are recovered to within the 68% credible interval.We are not interested in an accuracy of recovery of theHOD parameters, and we will not pay much attention tothe HOD parameters.

B. Validation strategy against analysis setups:scale cuts, parameter degeneracies, observational

effects and RSD

An advantage of the emulator based halo model is thatDark Emulator gives an accurate prediction of thehalo correlation functions (ξhm and ξhh) including all thenonlinear effects down to small scales (nonlinear clus-tering, nonlinear bias, and the halo exclusion effect) andtheir dependences on cosmological models within wCDMframework. To evaluate the performance of the emulatorbased method, we study various setups as summarized inTable V.

One question we want to address in this paper is; whichscale cuts for ∆Σ and wp are adequate in parameter esti-mation? Since ∆Σ and wp have higher signal-to-noise ra-tios at smaller scales, we want to include the informationof ∆Σ and wp down to smaller scales in the nonlinear 1-halo term regime. However, such smaller scales are moreaffected by nonlinear physics, especially galaxy physics,so it would be difficult to accurately model the cluster-ing signals on very small scales. Including such small-scale information might cause a bias in the estimatedcosmological parameters. We should avoid such a failuresituation as much as possible. To estimate appropriatescale cuts, we will study the performance of the methodadopting different scale cuts of (0.5, 0.75), (1,1.5), (2, 3)or (8,12) (in units of h−1Mpc) for wp and ∆Σ, respec-tively, where (2, 3) is our fiducial choice unless explicitlystated. For all the cases we adopt Rmax = 30 h−1Mpcfor the maximum scale up to which we include the in-formation of ∆Σ and wp for parameter estimation. Thuswe do not include the BAO information for all the anal-yses. To study the impact of the maximum scale, wealso study the setup for Rmax = 70 h−1Mpc, fixing otherparameters to those in the fiducial setups.

For the setups labeled as “∆Σ alone” and “wp alone”,we study the parameter constraints if using either of ∆Σor wp alone. Comparing this result with the baselinemethod manifests complementarity of ∆Σ and wp in thecosmological parameter estimation.

To study the impact of the observational effects onparameter estimation, we include the geometrical depen-dence of Ωm and introduce additional parameter to model

18

TABLE V. A summary of the analysis setups.

setup scale cut sample parameters Note[h−1Mpc]

baseline (2, 3) (Ωm, σ8)+HOD (17 paras.) fiducial mock (Rmax = 30 h−1Mpc)+RSD (2,3) – include RSD effect in the fiducial mockscale cuts (0.5, 0.75) – –

(1, 1.5) – –(8, 12) – –

Rmax = 70 h−1Mpc (2,3) – use up to Rmax = 70 h−1Mpc∆Σ alone (2,3) – –wp alone (2,3) – –Ωm-goem. (2,3) – via (Σcr, R)photo-z error (∆zph) (2,3) +∆zph only in modelshear-m (∆mγ) (2,3) + ∆mγ only in modelΩm,geom+∆zph+∆mγ (2,3) + (Ωm,geom,∆zph,∆mγ) only in modelfull HSC (2,3) – use 0.1× Cov∆Σ mimicking the full HSC data

the photo-z error and/or multiplicative shear bias, as dis-cussed in Sections II D 1–II D 3.

For the setup labeled as “RSD”, we study the impactof RSD effect. In the theoretical template we model theRSD effect using the linear RSD model. Then we com-pare the theoretical template with the mock catalog in-cluding the full RSD effects, and then assess whether thetheoretical model is still applicable to a realistic setup,without any significant bias or degradation in the esti-mated parameter.

Finally, we show a forecast of how the anticipated fullHSC dataset covering 1400 sq. deg., about factor of 10larger area than the HSC-Y1 data, can improve the cos-mological constraints. To do this forecast, we simplyscales the covariance of ∆Σ by the area factor betweenthe HSC-Y1 and full datasets.

C. Validation strategy against uncertainties inhalo-galaxy connection

The HOD model is an empirical prescription of thehalo-galaxy connection. Our expectation is that we couldrecover the underlying cosmological parameters as longas a sufficient number of halo-galaxy connection parame-ters are introduced and then the effects are marginalizedover when estimating cosmological parameters. To as-sess the “robustness” of our emulator based halo modelagainst uncertainties in halo-galaxy connection, we use awide variety of mock galaxy catalogs to study whetherthe baseline method can recover the true cosmologicalparameters for the different catalogs, as summarized inTable VI.

Fig. 5 compares the mock signals of ∆Σ and wp fordifferent mocks relative to those of the fiducial mock.Here all the mocks, except for the “cent-incomp.” and“FoF-halo” mocks, are built using the same HOD asthat of the fiducial mocks, but using different ways of

populating galaxies into individual halos, as describedbelow. The variety in the target observables for a fixedHOD is possible, because the HOD only specifies the av-erage number of galaxies per halo in each mass bin, butthere are many different ways to populate galaxies intohalos leading to different spatial distributions (cluster-ing properties) of galaxies, even for the same HOD. Thedifferent mock catalogs lead to modifications in the am-plitudes and scale-dependences of ∆Σ and wp in a com-plex way, compared to the fiducial mock. One of themost important systematic effects is the assembly biaseffect, and we also use the mock catalogs, labeled as“assembly-b-ext” and “assembly-b”, to test the per-formance of our method against the assembly bias effect.

In the following we describe details of each mock.Readers, who are interested in the results, can skip thissection and directly go to Section V.

1. Satellite galaxies

Even if the HOD model is fixed, there are several waysof populating satellites in halos in each simulation real-ization. To study the impact of variations in the distri-bution of satellite galaxies, we construct several mocks,for the same HOD as that of the fiducial mock.

The “sat-mod” mock is a slight modification from thefiducial mock. In this mock we populate satellite galax-ies in halos irrespective of whether each halo alreadyhosts a central galaxy. In this mock there are haloswhich host only satellite galaxy(ies) inside, without acentral galaxy. Here we assume that the radial distri-bution of satellite galaxies follows the NFW profile as inthe fiducial mock.

For the “sat-DM” mock, we populate satellitegalaxy(ies) in each host halo by randomly assigning eachsatellite to dark matter particles in the halo, in contrastto the NFW profile.

19

TABLE VI. A summary of the mock catalogs we use in this paper to assess “robustness” of the baseline method againstvariations in halo-galaxy connection. All the mock catalogs, except for “cent-imcomp.” and “FoF-halo” catalogs, have thesame HOD in average sense, but leads to modifications in ∆Σ and wp in their amplitudes and scale-dependences. The column“satellite gals.” denotes a model of the spatial distribution of satellite galaxies in the host halo. In the columns of ∆Σ and wp,“X” or “–” denote whether they are modified from the fiducial mock or not, respectively.

Model HOD satellite gals. ∆Σ wp descriptionfiducial fid. NFW – – fiducial modelRSD fid. NFW – X include the RSD effect in wsat-mod fid. NFW X X populate satellites irrespectively of centralssat-DM fid. DM part. X X populate satellites according to N -body particlessat-sub fid. subhalos X X populate satellites according to subhalosoff-cent1 fid. NFW X X all centrals off-centered, with Gaussian profileoff-cent2 fid. NFW X X a fraction (0.34) of “off-centered” centrals, assuming Gaussian profileoff-cent3 fid. NFW X X similar to “off-cent1”, but with NFW profileoff-cent4 fid. NFW X X similar to “off-cent2”, but with NFW profilecent-incomp. 〈Nc〉 mod. NFW X X include an “incomplete” selection of centralsFoF-halo mod. FoF halos X X use FoF halos to populate galaxiesassembly-b-ext fid. NFW X X populate galaxies according to concentrations of host halosassembly-b fid. NFW X X similar to “assembly-b-ext”, but introduce scattersbaryon fid. NFW X – mimic the baryonic effect of Illustris on the halo mass profile

For the “sat-subhalo” mock, we populate satellitegalaxy(ies) in each host halo by randomly assigning eachsatellite to subhalo(s) in the host halo, which are takenfrom the Rockstar output.

2. Off-centering effects of “central” galaxies

For the fiducial mocks, we assume that “central”galaxies are located at the center (the highest mass den-sity) of each host halo. However, a central galaxy in ahost halo can be “off-centered” as a result of merger or ac-cretion in a hierarchical structure formation [54, 55]. Tomimic this possible effect, we generate mock catalogs in-cluding the off-centering effects of central galaxy in eachhalo. Note here that we mean, by “central” galaxies,galaxies that are populated into halos according to thecentral HOD, and the central galaxies can be off-centeredfrom the true halo center. More physically speaking, agalaxy which resides in the most massive subhalo can beconsidered as a central galaxy, but the galaxy can be off-centered due to the physical effects we discussed above.

We generate four kinds of mock catalogs including theoff-centering effects, following the method in Kobayashiet al. [48] [also see 21]. The off-cent1 mock is de-signed to include the maximum possible amount of off-centering effect, where we assume that all central galax-ies are off-centered from the true halo center of each hosthalo. We assume that the average radial profile of off-centered galaxies with respect to the halo center followsa Gaussian profile with width Roff = 2.2, i.e. given bypoff(k) ∝ exp

[−k2(Roffrs)

2/2]

for the radial profile inFourier space, where Roff is a parameter to characterizethe typical off-centering radius relative to the scale radius(rs) of the NFW profile. When we populate a central

galaxy in each halo, we randomly draw an off-centeringradius from the Gaussian profile, and then place thegalaxy into the spherical shell of the off-centering radiusin the host halo (with randomly choosing the angular po-sition for the azimuthal angles). Then we populate satel-lite galaxies in the same way as that for the fiducialmock.

For the “off-cent2” mock, we assume that a frac-tion of central galaxies are off-centered. Following theimplication found in More et al. [25] [54] for the SDSSgalaxies, we assume qoff = 0.34 of the central galaxies areoff-centered, while the remaining galaxies (1−qoff = 0.66)are at the halo center. We then populate satellite galaxiesin the same way as that for the fiducial mock.

The “off-cent3” mock is very similar to theoff-cent1 mock, but we populate the off-centered “cen-tral” galaxies into halos assuming that the off-centeredgalaxies follow the NFW profile of the host halo, similarlyto satellite galaxies.

The “off-cent4” mock is very similar to theoff-cent2 mock, but we populate the off-centered galax-ies assuming the NFW profile as in off-cent3 mock.

3. FoF halos

Dark matter halos are neither uniquely-defined ob-jects nor have a clear boundary with the surroundingstructure. In this paper we use those identified by theRockstar algorithm with the spherical-oversensity (SO)masses in each simulation realizations as our defaultchoice (see Section III A). For the “FoF halo” mock,we use halos that are identified by the friends-of-friends(FoF) method with linking length bFoF = 0.2, in simula-tion realizations. The FoF halos do not necessarily have

20

0.50

0.75

1.00

1.25

1.50∆

Σ/∆

Σfid

LOWZ CMASS1 CMASS2

0.50

0.75

1.00

1.25

1.50

wp/w

fid p

sat-mod

sat-DM

sat-sub

off-cent1

off-cent2

off-cent3

off-cent4

0.50

0.75

1.00

1.25

1.50

∆Σ/∆

Σfid

100 101 102

R [h−1Mpc]

0.50

0.75

1.00

1.25

1.50

wp/w

fid p

100 101 102

R [h−1Mpc]100 101 102

R [h−1Mpc]

baryon

assembly-b-ext

assembly-b

cent-incomp.

FoF-halo

FIG. 5. The mock signals of ∆Σ and wp relative to those for the fiducial mock, for each of variants of the mock catalogs inTable VI. The error bars are the same as Fig. 2.

1012 1013 1014 1015

M [h 1 M ]

0.5

1.0

1.5

2.0

2.5

(dn F

oF/d

M)/

(dn S

O/d

M)

z = 0.251z = 0.484z = 0.617

FIG. 6. The ratio of the halo mass function of FoF halos tothat of SO halos in each mass bin. The data points show theresults at three redshifts z = 0.251, 0.484 and 0.617, corre-sponding to the representative redshifts of LOWZ, CMASS1and CMASS2, respectively. The data point in each bin is thethe mean of the ratios among 19 realizations of 1 (h−1Gpc)-size box simulations, and the error bar is the error on the meanamong 19 realizations, which is computed from the scattersamong 19 realizations, divided by

√19.

a one-to-one correspondence to the fiducial halos. Themass of each FoF halo is different from the SO mass evenif the corresponding halos are identified, because theirboundaries are different. Fig. 6 compares the mass func-tion of halos measured using the SO halo mass definitionor the FoF definition from the same N -body simulationrealization, at three redshifts corresponding to those forLOWZ, CMASS1, and CMASS2, respectively. The fig-ure shows a sizable difference in the halo mass functionsover the range of mass scales we consider.

To generate the “FoF-halo” mock we treat the FoFhalo mass of individual halos, as the mass argument forthe mean HOD functions, and then populate galaxies inFoF halos using the same HOD method as that for thefiducial mock.

4. Assembly bias effect

The “assembly bias” effect refers to the fact that theclustering amplitudes of galaxies or halos at large scalesdepends on a secondary parameter other than the halomass, especially depending on the assembly history ofgalaxies/halos [83–85]. The assembly bias is one of themost important physical effects causing a violation of thesimple halo model picture which assumes that clusteringproperties of halos are determined solely by halo mass.

21

To study whether or not cosmology inference based onour method is robust against the assembly bias effect,we use the mocks generated following the method inKobayashi et al. [43]. This is one of the most importanttests we address in this paper.

The “assembly-b-ext” mock is intended to study theworst case scenario for the assembly bias effect. To makethis mock, we first, for each halo, calculate the fraction ofmass enclosed within a sphere of 50% of the halo radiusR200 to the whole halo mass M200. We denote this innermass fraction as fin. We use this quantity as a proxyof the halo concentration, i.e., the higher fin means thehigher concentration. Then we make a ranked list ofhalos in which we sort the halo by ascending order of fin,in each narrow bin of halo masses. We populate centralgalaxies, according to the central HOD, into halos fromthe top of the list (from the lowest-concentration halo)in each mass bin. We then populate satellite galaxies inhalos that already host central galaxies using the satelliteHOD. For the mock generated in this way, we can havea maximum effect of the assembly bias in the large-scaleclustering amplitudes for all the three galaxy samples. Ascan be found from Fig. 5, wp(R) has larger amplitudesat large separations, by up to a factor of 1.6 than that ofwp in the fiducial mock, which is quite substantial.

For the mock assembly-b, we introduce a scatter tofin of each halo:

log10 fscatterin = log10 fin + ε. (36)

We assign a random scatter, ε, to each halo drawn froma zero-mean Gaussian distribution with σ, where σ isa parameter to control the amount of the scatter. Weadopt σ = 0.1 to generate the assembly-b mock for allthe three galaxy samples, which still leads to a significantboost in the clustering amplitudes in wp by up to a factorof 1.3 (therefore about halved strength compared to theassembly-b-ext mock) than that of wp in the fiducialmock. This might be more realistic for host halos of theSDSS galaxies (∼ 1013 h−1M), although the assemblybias effect has not been detected at a high significancefrom the real data. Note that the mean HOD is notmodified from the fiducial mocks with these procedures.

5. Baryonic effect

The baryonic effects inherent in galaxy forma-tion/evolution, which we are missing in our N -body sim-ulations, are another important physical systematic ef-fect, and we need to quantify their impact on the param-eter inference in our cosmology challenges. Although thebaryonic effects are still difficult to accurately model fromthe first principles, one should keep in mind some con-servation properties in the distribution of galaxies. First,massive galaxies like the SDSS LOWZ/CMASS galaxiesare likely to form at the same peaks of primordial den-sity fluctuations even in the presence of baryonic physics.Hence the distribution of massive galaxies relative to the

10−1 100 101

R [h−1Mpc]

0.6

0.8

1.0

1.2

∆Σ

(Bar

yon

)/∆

Σ(D

M−

only

)

Model

Leauthaud+17 Fig12

FIG. 7. We use the method in Schneider and Teyssier [45] tomodel the baryonic effect on the galaxy-galaxy weak lensingprofile, ∆Σ, where the mass conservation around halos hostinggalaxies is explicitly imposed. We tune the model parametersto reproduce the baryonic effect that is seen in the Illus-tris hydrosimulation [18]. Shown is the ratio of ∆Σ for thematched host halos of SDSS-like galaxies in the simulationswith and without the baryonic effect, which is taken fromFig. 12 in Leauthaud et al. [86]. Our method nicely capturethe baryonic effect, but we note that this would be a worstcase of the baryonic effect, because the Illustris (not Illus-trisTNG) employed the too large baryonic effect (especiallyAGN feedback).

total matter distribution, i.e. the bias function, is notlargely changed by baryonic physics [19, 87]. On theother hand, the baryonic physics causes a redistributionof matter around each galaxy, e.g. due to various effectssuch as dissipative contraction and supernova/AGN feed-backs. For this reason, the radial profile of matter distri-bution around a galaxy would likely be modified.

We follow the method in Schneider and Teyssier [45][also see 46] to include baryonic effects to the mock sig-nals. The notable feature of this model is that the modelexplicitly imposes the mass conservation around halos,and models the baryonic effects as a redistribution of thesurrounding matter around each halo (more exactly thehalo profile). We tuned the model parameters so that themodel prediction reproduces the baryonic effect on thelensing profile, ∆Σ, in the original Illustris hydrosim-ulation [18] that implemented too large baryonic effectthan implied by observations, as explicitly demonstratedin Fig. 7. Hence the baryon mocks are considered as aworst case of the baryonic effect on the weak lensing pro-file. For wp, we use the mock signal for the fiducialmocks; that is, we do not include a possible effect of thebaryonic physics on the distribution of galaxies in thehost halo.

22

100 101

r [h−1Mpc]

0.9

1.0

1.1

1.2r(ξ

)cc

(r)

LOWZ

100 101

r [h−1Mpc]

CMASS1

100 101

r [h−1Mpc]

CMASS2baseline

sat-mod

sat-DM

sat-sub

off-cent1

off-cent2

off-cent3

off-cent4

assembly-b-ext

assembly-b

cent-incomp.

FoF-halo

FIG. 8. Cross correlation coefficients, defined as r(ξ)cc (r) ≡ ξgm(r)/

√ξgg(r)ξmm(r), for different mock catalogs of SDSS-like

galaxies we use in this paper.

D. Cross-correlation coefficients in the mocks

All the mock catalogs we use in this paper have aproperty that the cross-correlation coefficient, rcc(r) ≡ξgm/[ξggξmm]1/2 ' 1 at the limit of large scales, greaterthan the size of massive halos, say at & 10 h−1Mpc,as shown in Fig. 8. This property is expected if galaxyphysics is confined to local, small scales, and becausethe clustering amplitudes at larger scales than the scaleof galaxy physics are governed by gravity alone (andproperties of primordial fluctuations) and then behaveas a linear biasing relation to the underlying matterdistribution at the large scales. Encouragingly this isconfirmed recently by using the hydrodynamical simu-lation IllustrisTNG in Ref. [58], where various galaxysamples, which are selected based on host halos andthe various environment parameters, display rcc ' 1 atr & 10 h−1Mpc. The Dark Emulator outputs also pre-dict r ≡ ξhm/[ξhhξmm]1/2 ' 1 for halo correlation func-tions at scales greater than the size of halos. However,even for the scales where rcc ' 1, this does not meanthat the linear theory serves as an accurate theoreticaltemplate [26]. It is important to include the nonlinearclustering, the nonlinear halo bias and the halo exclusioneffect in the theoretical template.

We also note that rcc can be greater than unity;rcc ≥ 1, which occurs in the 1-halo term regime wherethe sub-Poisson nature 〈Ng(Ng − 1)〉 6= 〈Ng〉2 becomesimportant due to a finite number statistics of galaxies inthe same host halo [25, 33, 36].

V. RESULTS

In this section we show the main results of this pa-per, which are assessment and validation of the emulatorbased halo model method for cosmology inference. Herewe mean by “validation” whether the method can re-cover the true cosmological parameters, Ωm0, σ8 and S8,to within 68% credible intervals, after marginalizationover the HOD parameters (galaxy-halo connection pa-

rameters). Note that we will not focus on the accuracyof recovering the HOD parameters.

A. Validation of the baseline method,complementarity of ∆Σ and wp, and tests with

different scale cuts

First we perform a sanity check; we study whetherour baseline method (see Table V) can recover the truecosmological parameters when comparing the theoreti-cal templates to the mock signals measured from thefiducial mock that is based on the same HOD modelused in the theoretical template. Different model pa-rameters affect the observables in a complex way, so itis not obvious whether the baseline method can recoverthe true cosmological parameters, after projecting theposterior distribution in a multi-dimensional parameterspace onto a sub-space including the cosmological param-eters. In fact, as discussed in Refs. [26, 82], the modesof the marginalized posterior distribution of cosmologicalparameters could be biased from the true values, if theparameters suffer from severe degeneracies.

Fig. 9 shows the results. Here we employ Rcut = 2 and3 h−1Mpc for the scale cuts for wp and ∆Σ above whichwe include the clustering and lensing information in pa-rameter inference (see Table V). The figure nicely showsthat the lensing and clustering information are comple-mentary to each other, and combining the two lifts theparameter degeneracies. Thus Fig. 9 gives a validationof the baseline method; the baseline method can recoverthe cosmological parameters if properties of the galaxyclustering for SDSS-like galaxies are close to the fiducialHOD model we employed. The baseline method achievesa precision of σ(S8) ' 0.035 for the SDSS and HSC-Y1data. In the following we show mainly the results forthe joint probes combining the information of ∆Σ andwp. For completeness of our discussion, in Fig. 22 weshow the posterior distribution of the parameters includ-ing all the HOD parameters for the CMASS1-like galaxysample.

23

0.2 0.4

Ωm0

0.2

0.4

0.6

0.8

σ2 lo

gM

(zC

MA

SS

1)

13

14

logM

min

(zC

MA

SS

1)

0.6

0.8

1.0

1.2

S8

0.6

0.8

1.0

1.2

σ8

0.6 0.8 1.0

σ8

0.6 1.0

S8

13 14

logMmin(zCMASS1)

0.2 0.5 0.8

σ2logM(zCMASS1)

lensing (∆Σ) alone

clustering (wgg) alone

baseline: ∆Σ + wgg

FIG. 9. Marginalized posterior distribution in each 2D sub-space of the parameters, obtained from the projected correlationfunction information alone (wp; orange-color contours), the lensing information alone (∆Σ; blue) and the joint constraints(green), respectively. The inner and outer contours show the 68 % and 95 % credible regions, respectively. We adopt thebaseline setup for which we employ Rcut = 2 and 3 h−1Mpc for the scales cuts of wp and ∆Σ, respectively. For the mocksignals, we use the signals measured from the fiducial mocks which are generated using the same HOD model as those in thetheoretical templates. Here we include 17 model parameters in the parameter inference: 2 cosmological parameters (Ωm andσ8) plus 5 HOD parameters for each of the 3 galaxy samples (LOWZ, CMASS1 and CMASS2). Here we show, as an example,the results for the central HOD parameters for the CMASS1 sample at z = 0.484, i.e. Mmin and σlogM . The vertical andhorizontal dashed lines denote the input parameter value used in the mock catalog.

What is the impact of HOD parameters on cosmolog-ical parameter estimation? To answer this question, inFig. 10 we show the results when fixing the HOD param-eters to their fiducial values. It is clear that the HODparameters cause significant degradations in the cosmo-logical parameters; for example, the marginalized error

of Ωm is enlarged by a factor of 3 when including theHOD parameters. However, this is a price one has topay to obtain robust constraints on cosmological param-eters when including the small-scale information. If oneuses a more aggressive method, e.g. by using a less flexi-ble model of halo-galaxy connection (such as fixing some

24

0.30 0.38

Ωm0

0.8

0.9

S8

0.7

0.8

0.9

1.0

σ8

0.8 1.0

σ8

0.8 0.9

S8

baseline

fixed HOD

FIG. 10. Similar to the previous figure, but shown are the re-sults when fixing the HOD parameters in the model templatesto their fiducial (input) values. The green contours (labeledas “baseline”) are the same as those in Fig. 9.

HOD parameters), one could suffer from severe biases incosmological parameters.

Our results might also be compared to a more conser-vative approach, e.g. a method using only the clusteringobservables at large scales, i.e. not including the small-scale information. In Fig. 11 we compare the results fromthe halo model method, studied in this paper, with thosefrom the perturbation theory (PT) based method studiedin Sugiyama et al. [26]. They employed the “minimal”-bias model using the fully nonlinear matter power spec-trum and the linear bias parameter (b1); ξgm = b1ξ

NLmm

and ξgg = b21ξNLmm, where the halofit model is used to

model the nonlinear ξNLmm. It was shown that, as long

as the conservative scale cuts Rcut = 12 and 8 h−1Mpcfor ∆Σ and wp are employed and the bias parameter istreated as a free parameter, the minimal bias methodpasses all the validation tests against a variety of themock catalogs including the assembly bias mock. Hence,the results for the minimal bias method can be consideredas conservative, yet robust parameter constraints thatcan be extracted from the SDSS and HSC-Y1 data. Thefigure clearly shows that including the small-scale infor-mation and the halo bias information can improve cosmo-logical constraints compared to the conservative method.In particular, the halo model based method gives a factorof 2 or 3 improvement in the marginalized error of σ8 orS8, respectively, compared to that for the minimal biasmethod. Thus the halo model method has a potentialto obtain the improved cosmological constraints, if themodel is flexible enough to describe variations in galaxy

0.3 0.4

Ωm0

0.6

0.8

1.0

S8

0.6

0.8

1.0

1.2

σ8

0.6 1.0

σ8

0.6 0.8 1.0

S8

minimal bias model

baseline

FIG. 11. Similar to Fig. 9, but the figure compares themarginalized posterior distributions obtained from the HODbased method, studied in this paper, and the perturbationtheory (PT) inspired method in Sugiyama et al. [26]. Herewe apply both the methods to the same fiducial mock sig-nals. For the PT method, we model the matter-galaxy cross-correlation (ξgm) and the galaxy auto-correlation (ξgg) by thenonlinear matter auto-correlation multiplied by a linear biasparameter: ξgm = b1ξ

NLmm and ξgg = b21ξ

NLmm. We then treat b1

as a free parameter in the parameter inference, and employthe scale cuts of Rcut = 12 and 8 h−1Mpc for ∆Σ and wp,respectively, compared to Rcut = 3 and 2 h−1Mpc for thehalo model based method. The green contours are the sameas those in Fig. 9.

clustering at small scales down to a few Mpc, which wewill test later.

In Fig. 12 we give a summary of the performancefor different setups. The second to the fourth row, ly-ing between the horizontal dashed lines, show the re-sults when using the different scale cuts, (R∆Σ

cut, Rwp

cut) =(0.5, 0.75), (1.0, 1.5) or (8, 12) h−1Mpc, instead of (2, 3)as our fiducial choice. For the fiducial mocks, thesesmaller scale cuts apparently recover the cosmologicalparameters; the true value of each parameter is withinthe 68% credible interval. However, a closer look alsoreveals that the size of the credible interval is not signif-icantly improved by including the smaller information.We also checked that, when applying the smaller scalecuts to other mock catalogs rather than the fiducialmocks, those lead to a larger bias in the cosmologicalparameters compared to the baseline method. Hence,with these results, we conclude that the fiducial scalecuts of (2, 3) h−1Mpc are reasonable. On the other hand,the row labeled as “Rmax = 70 h−1Mpc” shows the re-sults when including the information of ∆Σ and wp up

25

0.8 0.9S8

0.7 0.8 0.9σ8

0.25 0.35Ωm0

baseline: (2,3)

scale cuts: (0.5, 0.75)

scale cuts: (1.0, 1.5)

scale cuts: (8, 12)

Rmax = 70h−1Mpc

lensing alone

wp alone

Ωm0-geom.

∆zph: 0.04

∆zph: 0.1

∆mγ: 0.01

Ωm0+∆zph+∆mγ: (0.1, 0.01)

+RSD

full HSC

minimal bias (Sugiyama+ 2020)

FIG. 12. Summary of the estimation of each cosmological parameter, Ωm, σ8 or S8(≡ σ8Ω0.5m ), for the different setups in

Table V. The blue dot in each row denotes the mode of the marginalized posterior distribution of each parameter, and the errorbar denotes the 68% credible interval, which is computed from the highest density interval of the marginalized 1D posteriordistribution. The vertical red line denotes the true value used in the mock catalog, and the shaded region denotes the 68%credible interval for the baseline setup for comparison.

to Rmax ' 70 h−1Mpc instead of our default choiceRmax = 30 h−1Mpc. Note that this maximum scale isstill below the BAO scales, and we do not include theBAO information. It is clear that the larger-scale infor-mation does not improve the cosmological parameter es-timation, and 30 h−1Mpc seems sufficient for the signal-to-noise level expected for the HSC-SDSS analysis.

B. Comparison with analytic halo model

In Fig. 13, we compare the results for our baselinemethod with those obtained by comparing the model pre-dictions of analytical halo model [88] in More et al. [25]with the mock signals from the fiducial mocks. Theanalytical halo model was constructed by calibrating thestandard halo model [33] with N -body simulation resultsavailable at that time, e.g. to include the halo exclusioneffect and the nonlinear halo bias effect [see Ref. 47, fordetails]. For example, the analytical halo model uses thefitting formula of halo bias developed in Ref. [89, 90],but several recent studies point out an inaccuracy in thefitting formula, up to 5% in the bias amplitude [see thebottom panel of Fig. 22 in Ref. 40] [also see Refs. 91–93]. For the analytical halo model, we did not includethe off-centering effect or incompleteness effect as in our

baseline method.The figure shows sizable differences in all the cosmo-

logical parameters between our emulator based methodand the analytical halo model. The 1D posterior dis-tribution shows that the analytical halo model cannotrecover the true values of σ8 and S8 to within the 68%credible interval. In addition, the degeneracy directionsin the projected two parameter subspace are quite differ-ent. Figuring out the cause of the difference is beyondthe scope of this paper, but here at least we would liketo stress that the emulator-based halo model displays abetter performance.

C. The impacts of observational effects: thegeometrical correction, photo-z errors, shear

multiplicative bias and RSD

As discussed in Section II D, various observational ef-fects could affect the cosmological inference from themeasured ∆Σ and wp. For the results discussed up tothe preceding subsections, we ignored these effects, andin this section we study the impact of these effects.

The rows with labels starting from “Ωm-geom.” to“+RSD” in Fig. 12 show the results when including eachor all of these observational effects. First let us discuss

26

0.3 0.4

Ωm0

0.8

0.9

1.0

1.1

S8

0.8

1.0

σ8

0.8 1.0

σ8

0.8 1.0

S8

baseline

analytic halo model

FIG. 13. Similar to Fig. 9, but the plot shows the results forthe baseline method with those obtained by using the analytichalo model in More et al. [25]. Here we used the same scalecuts, Rcut = 2 and 3 h−1Mpc, for wp and ∆Σ, respectively,for both methods.

the results for “+RSD”. For this test we use the mock cat-alogs where we use the fiducial HOD to populate galaxiesinto the simulation realizations and include the RSD ef-fects due to the peculiar velocities of individual galaxies[43, 48]. Then we perform the cosmology inference bycomparing the theoretical templates, including the linearKaiser effect (Section II D 4), with the mock signals. Thefigure shows that the linear Kaiser model can properlytake into account the RSD effect for our fiducial pro-jection length (πmax = 100 h−1Mpc), and there is nodegradation in the parameter estimation. This is similarto the result in Sugiyama et al. [26]. Although the RSDeffect itself carries the dependence on Ωm (more exactlyvia the growth rate), it leads to only slight improvementin Ωm.

Now we discuss the results for “Ωm-geom.”, whichrefers to the fact that a reference cosmology needs tobe assumed to measure ∆Σ(R) and wp(R) from directobservables, and the assumed cosmology generally dif-fers from the underlying true cosmology. Exactly speak-ing, for a flat-geometry ΛCDM model, we have to “re-measure” ∆Σ(R) and wp(R) every time Ωm is varied inparameter estimation. This might be time-consuming,and indeed More [60] showed that this conversion canbe safely done by a multiplicative factor taking into ac-count the geometrical dependence, which we employ inthis paper. Fig. 12 shows that including the Ωm geo-metrical dependence does not cause a bias in the cosmo-logical parameters. However, a closer look shows that

including this dependence slightly enlarges the credibleinterval. This happens as follows. If we assume a slightlylarger value of Ωm, it leads to the smaller amplitude inthe model prediction of wp (recall that the current con-straint is mainly from the wp information). However,when assuming such a model with increased Ωm in themeasurements, it leads to the smaller amplitudes in themeasured wp as shown in Fig. 2. Thus the dependencesof Ωm are compensated to some extent in the model andthe measurement. This is the reason that the credibleinterval of Ωm is slightly degraded.

Next we discuss the impacts of photo-z errors and themultiplicative shear bias, which are among the most im-portant systematic errors in the weak lensing measure-ments. As discussed in Sections II D 2 and II D 3, we in-troduce nuisance parameters, denoted as ∆zph and ∆mγ ,to model these effects. To do this we employ the parame-ters to model the systematic effects that could be presentfor the actual HSC data. For a sample of source galax-ies, we assume that source galaxies are selected basedon their photo-z’s satisfying a conservative cut that thephoto-z posterior distribution of individual galaxies sat-isfies

∫∞0.75

dz p(z) ≥ 0.99 [94, 95], where the lower cutoffof the integration, z = 0.75, is well above the redshifts ofCMASS galaxies (the maximum redshift cut of CMASSgalaxies is 0.7 as shown in Table II). After using the re-weighting method in [4] to infer the intrinsic redshift dis-tribution of source galaxies, we compute the average ofthe lensing critical surface density over the source dis-tribution,

⟨Σ−1

cr

⟩, that is used in the ∆Σ measurement

for each sample of lensing galaxies (LOWZ, CMASS1and CMASS2). Then, by shifting the posterior distri-bution of all the source galaxies by the same amount,∆zph, we repeat the same calculation of

⟨Σ−1

cr

⟩to esti-

mate a shift in the ∆Σ signal. After computing thesequantities for multiple values of ∆zph, we include ∆zph

to compute the shift in ∆Σ from an interpolation of thepre-computed values of

⟨Σ−1

cr

⟩. In this paper we study

the impact of photo-z errors on parameter estimation byadopting the Gaussian prior with width σ(∆zph) = 0.04or 0.1. Here σ(zph) = 0.04 corresponds to the quotederrors of photo-z estimation for the source galaxies, es-timated using the same method in [4]. We also adopt avery conservative prior of σ(∆zph) = 0.1, which is abouta factor of 2.5 larger than the quoted errors. Fig. 12shows that the cosmological parameters remain almostunchanged by the photo-z errors. Thus, as claimed inOguri and Takada [21], a self-calibration of the photo-zerrors seems to work very well, by using the single sampleof source galaxies and utilizing the dependence of lensingefficiency on lens redshifts.

For a multiplicative shear bias, ∆mγ , we employ theGaussian prior σ∆mγ

= 0.01 as recommended based ondedicated image simulations of HSC galaxies in Mandel-baum et al. [42, 96]. Fig. 12 shows that the multiplica-tive shear bias does not either affect the cosmologicalparameters. Thus, the method of Oguri and Takada[21] allows us to self-calibrate both the photo-z errors

27

and the multiplicative bias errors. The row denoted as“Ωm +∆zph +∆mγ” shows the results including all thesethree effects. The results are not so different from thebaseline method ignoring these effects. Hence we con-clude that these observational effects are not a severesource of systematic errors causing a sizable bias in thecosmological parameters.

When including these observational effects, the base-line method can achieve a precision of σ(S8) ' 0.042.

D. Tests and validations against uncertainties inhalo-galaxy connection

In this section, to assess the performance of the halomodel method against uncertainties in the halo-galaxyconnection, we perform the parameter inference againstvarious mock catalogs (Table VI). Note that we hereemploy the fiducial HOD model (5 parameters model),which does not include the off-centering effect, the in-completeness effect, the baryonic effect nor the assemblybias effect. Here we address whether the fiducial halomodel can recover the true cosmological parameters towithin the credible intervals after marginalizing over theHOD parameters. Even if we use a weak prior on thenumber density of galaxies (the abundance), e.g. 50% or5% of the number density, the following results remainalmost unchanged.

Fig. 14 gives a summary of the results. The figureshows that, except for the assembly bias mocks (and theextreme off-centering mock, off-cent1, for Ωm0 and σ8),the halo model method with the baseline setup recov-ers the input cosmological parameters to within the 68%credible interval. We should also note that S8 is betterrecovered compared to Ωm0 or σ8. This is an encourag-ing result, because S8 is close to the primary parametercombination that determines the amplitudes of wp and∆Σ. We below discuss the results for some variants ofthe mocks in more detail.

Figs. 15 and 16 show the marginalized distributions ofthe cosmological parameters for the off-cent, FoF-haloand cent-incomp. mocks (see Table VI). For most casesexcept for the off-cent1 mock, where all the centralgalaxies are off-centered, the halo model method recov-ers the true cosmological parameters within 68% credibleinterval. As shown in Fig. 4, the off-cent1 mock givessmaller amplitudes in ∆Σ at scales around the scale cut(Rcut = 3 h−1Mpc), but does not change the wp ampli-tude at scales greater than the scale cut (R = 2 h−1Mpc).The smaller lensing amplitude leads to an underestima-tion in the average mass of host halos. Since smaller-masshalos give a smaller bias amplitude for a fixed cosmology,this would lead to a lower σ8 to reproduce the amplitudeof wp in the mock signal (see Fig. 2). This explains a neg-ative bias in σ8 for the off-cent1 mock. We confirmedthat, if we employ the larger scale cut for ∆Σ, the biasin σ8 is mitigated. For the FoF-halo mock, our methodnicely recovers S8, although we find sizable biases in Ωm0

and σ8.In Fig. 17, we show the results for the baryon,

assembly-b and assembly-b-ext mocks. The figureshows that the assembly bias mocks lead to biases inthe cosmological parameters, greater than the credibleinterval. Thus the assembly bias is indeed the most dan-gerous systematic effect, which violates the scaling rela-tion of halo bias with halo mass. However, we note that,in this paper, the result is considered as the worst-casescenario, because we include the maximum possible ef-fect of the assembly bias in the assembly-b-ext mock,where we populate galaxies into halos assuming a fullydeterministic assignment of centrals in ascending order ofhalo concentration: we populate galaxies from the lowestconcentration halos in each mass bin. The impact of theassembly bias for more realistic galaxies, even if existsfor SDSS galaxies, would be much smaller than what isshown in this paper. Nevertheless, this possible impactof the assembly bias needs to be kept in mind.

To tackle the possibility that the actual galaxy sampleis suffering from the assembly bias effect, we here proposea practical solution to assess its impact or mitigate it inparameter estimation. In Fig. 18, we show the resultswhen using different scale cuts, (4, 6) or (8, 12) h−1Mpc,for wp and ∆Σ, instead of our fiducial choice (2, 3). Notethat (8, 12) is the choice of the minimal bias method inSugiyama et al. [26] (also see Figs. 11 and 14). The figureclearly shows that the posterior distribution systemati-cally moves with varying the scale cuts, and then thechoice of (8, 12) recovers the true cosmological parame-ters. In this case, the method does not use the 1-haloterm signal of ∆Σ, and tries to fit the mock signals bythe model templates, where rcc ' 1 is satisfied. Thus,if we observe a similar systematic shift in the parameterconstraints with varying the scale cuts, we could identifya signature of the assembly bias effect in actual data.

VI. DISCUSSION AND CONCLUSION

In this paper we have in detail studied the performanceof the halo model based method for cosmological param-eter estimation. We used Dark Emulator to model thehalo clustering quantities (halo mass function, halo auto-correlation function and halo-matter cross correlation),where the emulator includes, by design, all non-trivial ef-fects such as nonlinear clustering, nonlinear halo bias andhalo exclusion effect that are otherwise difficult to analyt-ically model. We combined Dark Emulator with theHOD method to model clustering observables of galaxiesfor which we consider the projected correlation function,wp(R), and the galaxy-galaxy weak lensing ∆Σ(R) in thispaper. Then we validate the emulator-based halo modelmethod by studying whether to recover the cosmologi-cal parameters from MCMC analyses by comparing themodel predictions with the mock signals for the spectro-scopic SDSS galaxies and the HSC-Y1 galaxies.

The main results of this paper are summarized as fol-

28

0.8 0.9S8

0.7 0.8 0.9σ8

0.25 0.35Ωm0

baseline

sat-mod

sat-DM

sat-sub

off-cent1

off-cent2

off-cent3

off-cent4

baryon

assembly-b-ext

assembly-b

assemnby-b-ext: (8,12)

assemnby-b: (8,12)

cent-incomp.

FoF-halo

FIG. 14. A summary of the performance of the baseline method against different mock catalogs. The circle shows the mode ofthe marginalized posterior distribution of each cosmological parameter, and the the error bar denotes the 68% credible interval.The vertical solid line denotes the true value of each parameter. The shaded region denotes the 68% credible interval for thefiducial mock, for comparison. The columns labeled as “assembly-b: (8,12)” and “assembly-b-ext: (8,12)” show the resultswhen the larger scale cuts of (8, 12) h−1Mpc are employed for wp and ∆Σ.

lows.

• Our method using Dark Emulator allows forcomputations of wp and ∆Σ at a few CPU sec-onds for each model, which is equivalent to a fac-tor of million reduction in computation time com-pared to the standard method (run N -body simula-tions, populate galaxies in halos, and then measurethe galaxy clustering observables from the mocks).With this emulator, we can perform the MCMCanalysis in practice.

• Changes in the cosmological parameters cause char-acteristic changes in the amplitudes and scale-dependences of wp and ∆Σ via a combination ofvarious effects: nonlinear clustering, nonlinear halobias and halo exclusion effect (Fig. 2).

• We showed that the halo model based method canrecover the underlying cosmological parameters, es-pecially S8 = σ8Ω0.5

m0, after marginalization overthe halo-galaxy connection (HOD) parameters, fora variety of mock catalogs except for the mocksincluding an extreme effect of assembly bias, for

the nominal choices of scale cuts, (Rwp

cut, R∆Σcut) =

(2, 3) h−1Mpc (Figs. 12 and 14). The baselinemethod can achieve a precision of σ(S8) ' 0.035–0.042 for the SDSS and HSC-Y1 datasets. Thismethod allows for tight constraints on the cosmo-logical parameter, because the small-scale informa-tion of ∆Σ(R) at scales, 3 ≤ R/[h−1Mpc] . 10, canbe used to infer the average mass of host halos ofthe SDSS galaxies, yielding useful information onthe galaxy bias that is sensitive to the large-scaleamplitudes of wp and ∆Σ.

• The halo model method suffers from sizable biasesin the cosmological parameters if the SDSS galaxieshave a significant assembly bias (although we con-sider the extreme mocks including the maximumamount of the assembly bias effect in this paper)(Figs. 14 and 17). Even if this is the case, weshowed that the cosmological parameters can berecovered if employing sufficiently large scales cutsof Rcut & 10 h−1Mpc, where the cross-correlationcoefficient rcc(r) ≡ ξgm(r)/[ξgg(r)ξmm(r)]1/2 ' 1(Fig. 18). This is equivalent to the case that we

29

0.3 0.4

Ωm0

0.8

0.9

S8

0.7

0.8

0.9

1.0

σ8

0.8 1.0

σ8

0.8 0.9

S8

baseline

off-cent1

off-cent2

off-cent3

off-cent4

FIG. 15. Marginalized posterior distributions of the cosmo-logical parameters when applying the halo model methodto the mock catalog where the off-centering effect of centralgalaxies is included (Table VI). For comparison, we show theresult for the fiducial mock, the same as the green contoursin Fig. 9.

0.3 0.4

Ωm0

0.8

0.9

S8

0.7

0.8

0.9

1.0

σ8

0.8 1.0

σ8

0.8 0.9

S8

baseline

FoF-halo

cent-incomp.

FIG. 16. Similar to the previous figure, but shown is theresult for the FoF and cent-incomp. mocks.

0.30 0.38

Ωm0

0.7

0.8

0.9

S8

0.7

0.8

0.9

1.0

σ8

0.8 1.0

σ8

0.7 0.8 0.9

S8

baseline

baryon

assembly-b

assembly-b-ext

FIG. 17. Similar to the previous figure, but shown is themarginalized posterior distributions for the baryon mocks andthe assembly bias mocks (assembly-b and assembly-b-ext).Here the assembly-b-ext is the worst (extreme) effect of as-sembly bias, where the large-scale amplitude of wp is modifiedby a factor of 1.6 (see Fig. 4). In the assembly-b mock, theboost in the amplitude is halved, which is still larger thanwhat is expected for an actual data, even if exists.

do not include the small-scale lensing informationin the cosmological parameter estimation. Howeverthe price to pay is that the statistical accuracy ofcosmological parameter determination is degraded,leading to σ(S8) ' 0.07 (see Fig. 14). Thus inpractice we should monitor whether the cosmolog-ical parameters have a systematic shift with differ-ent scale cuts, as a test of the assembly bias effectinherent in actual data (Fig. 17).

• We used the method using a single population ofsource galaxies to measure the galaxy-galaxy weaklensing for multiple lens galaxies at different red-shifts (LOWZ, CMASS1 and CMASS2) over therange of redshifts, z = [0.15, 0.7]. This method al-lows for self-calibration of the photometric redshifterrors and the multiplicative shear bias (Fig. 12).

Hence we conclude that we can safely apply the halomodel based method to actual SDSS and HSC-Y1 data.The cosmological constraints from actual observationaldata are presented in a companion paper (Miyatake etal. in prep.).

30

0.3 0.4

Ωm0

0.7

0.8

0.9

1.0

S8

0.8

1.0

σ8

0.8 1.0

σ8

0.7 0.9

S8

assembly-b-ext (2.0, 3.0)

assembly-b-ext (4.0, 6.0)

assembly-b-ext (8.0, 12.0)

FIG. 18. Similar to the previous figure, but we here studyhow biases in the cosmological parameters can be mitigatedby using the different scale cuts, (4, 6) or (8, 12) h−1Mpc forwp and ∆Σ, respectively, instead of the baseline setup (2, 3).The true cosmological parameters are recovered if using thelarge-scale cuts such as (8, 12); that is, if we do not includethe 1-halo term contribution.

ACKNOWLEDGMENTS

We would like to thank Ryoma Murata for his con-tribution during the early phase of this work. Thiswork was supported in part by World Premier Inter-national Research Center Initiative (WPI Initiative),MEXT, Japan, by JSPS KAKENHI Grant NumbersJP15H03654, JP15H05887, JP15H05893, JP15H05896,JP15K21733, JP17H01131, JP17K14273, JP18H04350,JP18H04358, JP19H00677, JP19K14767, JP20H01932,JP20H04723, JP20H05855, and JP20A203, by Japan Sci-ence and Technology Agency (JST) AIP AccelerationResearch Grant Number JP20317829, Japan. SS is sup-ported by International Graduate Program for Excellencein Earth-Space Science (IGPEES), World-leading Inno-vative Graduate Study (WINGS) Program, the Univer-sity of Tokyo. KO is supported by JSPS Overseas Re-search Fellowships.

[1] D. H. Weinberg, M. J. Mortonson, D. J. Eisenstein,C. Hirata, A. G. Riess, and E. Rozo, Phys. Rep. 530,87 (2013), arXiv:1201.2434.

[2] https://hsc.mtk.nao.ac.jp/ssp/.[3] H. Aihara, N. Arimoto, R. Armstrong, S. Arnouts, N. A.

Bahcall, S. Bickerton, J. Bosch, K. Bundy, P. L. Ca-pak, J. H. H. Chan, M. Chiba, J. Coupon, E. Egami,M. Enoki, F. Finet, H. Fujimori, S. Fujimoto, H. Furu-sawa, J. Furusawa, T. Goto, A. Goulding, J. P. Greco,J. E. Greene, J. E. Gunn, T. Hamana, Y. Harikane,Y. Hashimoto, T. Hattori, M. Hayashi, Y. Hayashi,K. G. He lminiak, R. Higuchi, C. Hikage, P. T. P. Ho,B.-C. Hsieh, K. Huang, S. Huang, H. Ikeda, M. Iman-ishi, A. K. Inoue, K. Iwasawa, I. Iwata, A. T. Jae-lani, H.-Y. Jian, Y. Kamata, H. Karoji, N. Kashikawa,N. Katayama, S. Kawanomoto, I. Kayo, J. Koda,M. Koike, T. Kojima, Y. Komiyama, A. Konno,S. Koshida, Y. Koyama, H. Kusakabe, A. Leauthaud,C.-H. Lee, L. Lin, Y.-T. Lin, R. H. Lupton, R. Man-delbaum, Y. Matsuoka, E. Medezinski, S. Mineo,S. Miyama, H. Miyatake, S. Miyazaki, R. Momose,A. More, S. More, Y. Moritani, T. J. Moriya, T. Mo-rokuma, S. Mukae, R. Murata, H. Murayama, T. Na-gao, F. Nakata, M. Niida, H. Niikura, A. J. Nishizawa,Y. Obuchi, M. Oguri, Y. Oishi, N. Okabe, S. Okamoto,Y. Okura, Y. Ono, M. Onodera, M. Onoue, K. Osato,M. Ouchi, P. A. Price, T.-S. Pyo, M. Sako, M. Saw-icki, T. Shibuya, K. Shimasaku, A. Shimono, M. Shi-

rasaki, J. D. Silverman, M. Simet, J. Speagle, D. N.Spergel, M. A. Strauss, Y. Sugahara, N. Sugiyama,Y. Suto, S. H. Suyu, N. Suzuki, P. J. Tait, M. Takada,T. Takata, N. Tamura, M. M. Tanaka, M. Tanaka,M. Tanaka, Y. Tanaka, T. Terai, Y. Terashima, Y. Toba,N. Tominaga, J. Toshikawa, E. L. Turner, T. Uchida,H. Uchiyama, K. Umetsu, F. Uraguchi, Y. Urata,T. Usuda, Y. Utsumi, S.-Y. Wang, W.-H. Wang, K. C.Wong, K. Yabe, Y. Yamada, H. Yamanoi, N. Yasuda,S. Yeh, A. Yonehara, and S. Yuma, PASJ 70, S4 (2018),arXiv:1704.05858 [astro-ph.IM].

[4] C. Hikage, M. Oguri, T. Hamana, S. More,R. Mandelbaum, M. Takada, F. Kohlinger, H. Miy-atake, A. J. Nishizawa, H. Aihara, R. Armstrong,J. Bosch, J. Coupon, A. Ducout, P. Ho, B.-C. Hsieh,Y. Komiyama, F. Lanusse, A. Leauthaud, R. H. Lup-ton, E. Medezinski, S. Mineo, S. Miyama, S. Miyazaki,R. Murata, H. Murayama, M. Shirasaki, C. Sifon,M. Simet, J. Speagle, D. N. Spergel, M. A. Strauss,N. Sugiyama, M. Tanaka, Y. Utsumi, S.-Y. Wang,and Y. Yamada, PASJ 71, 43 (2019), arXiv:1809.09148[astro-ph.CO].

[5] T. Hamana, M. Shirasaki, S. Miyazaki, C. Hik-age, M. Oguri, S. More, R. Armstrong, A. Leau-thaud, R. Mandelbaum, H. Miyatake, A. J. Nishizawa,M. Simet, M. Takada, H. Aihara, J. Bosch,Y. Komiyama, R. Lupton, H. Murayama, M. A. Strauss,and M. Tanaka, PASJ 72, 16 (2020), arXiv:1906.06041

31

[astro-ph.CO].[6] https://www.darkenergysurvey.org.[7] T. M. C. Abbott, F. B. Abdalla, A. Alarcon, J. Aleksic,

S. Allam, S. Allen, A. Amara, J. Annis, J. Asorey,S. Avila, D. Bacon, E. Balbinot, M. Banerji, N. Banik,W. Barkhouse, M. Baumer, E. Baxter, K. Bechtol,M. R. Becker, A. Benoit-Levy, B. A. Benson, G. M.Bernstein, E. Bertin, J. Blazek, S. L. Bridle, D. Brooks,D. Brout, E. Buckley-Geer, D. L. Burke, M. T. Busha,A. Campos, D. Capozzi, A. Carnero Rosell, M. Car-rasco Kind, J. Carretero, F. J. Castander, R. Cawthon,C. Chang, N. Chen, M. Childress, A. Choi, C. Con-selice, R. Crittenden, M. Crocce, C. E. Cunha, C. B.D’Andrea, L. N. da Costa, R. Das, T. M. Davis,C. Davis, J. De Vicente, D. L. DePoy, J. DeRose, S. De-sai, H. T. Diehl, J. P. Dietrich, S. Dodelson, P. Doel,A. Drlica-Wagner, T. F. Eifler, A. E. Elliott, F. El-sner, J. Elvin-Poole, J. Estrada, A. E. Evrard, Y. Fang,E. Fernandez, A. Ferte, D. A. Finley, B. Flaugher,P. Fosalba, O. Friedrich, J. Frieman, J. Garcıa-Bellido,M. Garcia-Fernandez, M. Gatti, E. Gaztanaga, D. W.Gerdes, T. Giannantonio, M. S. S. Gill, K. Glaze-brook, D. A. Goldstein, D. Gruen, R. A. Gruendl,J. Gschwend, G. Gutierrez, S. Hamilton, W. G. Hart-ley, S. R. Hinton, K. Honscheid, B. Hoyle, D. Huterer,B. Jain, D. J. James, M. Jarvis, T. Jeltema, M. D.Johnson, M. W. G. Johnson, T. Kacprzak, S. Kent,A. G. Kim, A. King, D. Kirk, N. Kokron, A. Ko-vacs, E. Krause, C. Krawiec, A. Kremin, K. Kuehn,S. Kuhlmann, N. Kuropatkin, F. Lacasa, O. Lahav,T. S. Li, A. R. Liddle, C. Lidman, M. Lima, H. Lin,N. MacCrann, M. A. G. Maia, M. Makler, M. Manera,M. March, J. L. Marshall, P. Martini, R. G. McMa-hon, P. Melchior, F. Menanteau, R. Miquel, V. Miranda,D. Mudd, J. Muir, A. Moller, E. Neilsen, R. C. Nichol,B. Nord, P. Nugent, R. L. C. Ogando, A. Palmese,J. Peacock, H. V. Peiris, J. Peoples, W. J. Percival,D. Petravick, A. A. Plazas, A. Porredon, J. Prat, A. Pu-jol, M. M. Rau, A. Refregier, P. M. Ricker, N. Roe,R. P. Rollins, A. K. Romer, A. Roodman, R. Rosen-feld, A. J. Ross, E. Rozo, E. S. Rykoff, M. Sako,A. I. Salvador, S. Samuroff, C. Sanchez, E. Sanchez,B. Santiago, V. Scarpine, R. Schindler, D. Scolnic,L. F. Secco, S. Serrano, I. Sevilla-Noarbe, E. Shel-don, R. C. Smith, M. Smith, J. Smith, M. Soares-Santos, F. Sobreira, E. Suchyta, G. Tarle, D. Thomas,M. A. Troxel, D. L. Tucker, B. E. Tucker, S. A. Ud-din, T. N. Varga, P. Vielzeuf, V. Vikram, A. K. Vi-vas, A. R. Walker, M. Wang, R. H. Wechsler, J. Weller,W. Wester, R. C. Wolf, B. Yanny, F. Yuan, A. Zen-teno, B. Zhang, Y. Zhang, J. Zuntz, and Dark EnergySurvey Collaboration, Phys. Rev. D 98, 043526 (2018),arXiv:1708.01530 [astro-ph.CO].

[8] C. Heymans, T. Troster, M. Asgari, C. Blake, H. Hilde-brandt, B. Joachimi, K. Kuijken, C.-A. Lin, A. G.Sanchez, J. L. van den Busch, A. H. Wright, A. Amon,M. Bilicki, J. de Jong, M. Crocce, A. Dvornik, T. Er-ben, F. Getman, B. Giblin, K. Glazebrook, H. Hoekstra,S. Joudaki, A. Kannawadi, C. Lidman, F. Kohlinger,L. Miller, N. R. Napolitano, D. Parkinson, P. Schneider,H. Shan, and C. Wolf, arXiv e-prints , arXiv:2007.15632(2020), arXiv:2007.15632 [astro-ph.CO].

[9] Y. Park and E. Rozo, Mon. Not. Roy. Astron. Soc.(2020), 10.1093/mnras/staa2647, arXiv:1907.05798

[astro-ph.CO].[10] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ash-

down, J. Aumont, C. Baccigalupi, M. Ballardini,A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak,R. Battye, K. Benabed, J. P. Bernard, M. Bersanelli,P. Bielewicz, J. J. Bock, J. R. Bond, J. Borrill,F. R. Bouchet, F. Boulanger, M. Bucher, C. Buri-gana, R. C. Butler, E. Calabrese, J. F. Cardoso,J. Carron, A. Challinor, H. C. Chiang, J. Chluba,L. P. L. Colombo, C. Combet, D. Contreras, B. P.Crill, F. Cuttaia, P. de Bernardis, G. de Zotti, J. De-labrouille, J. M. Delouis, E. Di Valentino, J. M.Diego, O. Dore, M. Douspis, A. Ducout, X. Dupac,S. Dusini, G. Efstathiou, F. Elsner, T. A. Enßlin,H. K. Eriksen, Y. Fantaye, M. Farhang, J. Fergusson,R. Fernandez-Cobos, F. Finelli, F. Forastieri, M. Frailis,A. A. Fraisse, E. Franceschi, A. Frolov, S. Galeotta,S. Galli, K. Ganga, R. T. Genova-Santos, M. Gerbino,T. Ghosh, J. Gonzalez-Nuevo, K. M. Gorski, S. Grat-ton, A. Gruppuso, J. E. Gudmundsson, J. Hamann,W. Handley, F. K. Hansen, D. Herranz, S. R. Hilde-brandt, E. Hivon, Z. Huang, A. H. Jaffe, W. C.Jones, A. Karakci, E. Keihanen, R. Keskitalo, K. Ki-iveri, J. Kim, T. S. Kisner, L. Knox, N. Krachmal-nicoff, M. Kunz, H. Kurki-Suonio, G. Lagache, J. M.Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence,M. Le Jeune, P. Lemos, J. Lesgourgues, F. Levrier,A. Lewis, M. Liguori, P. B. Lilje, M. Lilley, V. Lind-holm, M. Lopez-Caniego, P. M. Lubin, Y. Z. Ma,J. F. Macıas-Perez, G. Maggio, D. Maino, N. Man-dolesi, A. Mangilli, A. Marcos-Caballero, M. Maris,P. G. Martin, M. Martinelli, E. Martınez-Gonzalez,S. Matarrese, N. Mauri, J. D. McEwen, P. R. Mein-hold, A. Melchiorri, A. Mennella, M. Migliaccio, M. Mil-lea, S. Mitra, M. A. Miville-Deschenes, D. Molinari,L. Montier, G. Morgante, A. Moss, P. Natoli, H. U.Nørgaard-Nielsen, L. Pagano, D. Paoletti, B. Partridge,G. Patanchon, H. V. Peiris, F. Perrotta, V. Pettorino,F. Piacentini, L. Polastri, G. Polenta, J. L. Puget,J. P. Rachen, M. Reinecke, M. Remazeilles, A. Renzi,G. Rocha, C. Rosset, G. Roudier, J. A. Rubino-Martın,B. Ruiz-Granados, L. Salvati, M. Sandri, M. Savelainen,D. Scott, E. P. S. Shellard, C. Sirignano, G. Sirri, L. D.Spencer, R. Sunyaev, A. S. Suur-Uski, J. A. Tauber,D. Tavagnacco, M. Tenti, L. Toffolatti, M. Tomasi,T. Trombetti, L. Valenziano, J. Valiviita, B. Van Tent,L. Vibert, P. Vielva, F. Villa, N. Vittorio, B. D. Wandelt, I. K. Wehus, M. White, S. D. M. White, A. Zac-chei, and A. Zonca, Astronomy & Astrophysics 641,A6 (2020), arXiv:1807.06209 [astro-ph.CO].

[11] https://pfs.ipmu.jp/index.html.[12] M. Takada, R. S. Ellis, M. Chiba, J. E. Greene, H. Ai-

hara, N. Arimoto, K. Bundy, J. Cohen, O. Dore,G. Graves, J. E. Gunn, T. Heckman, C. M. Hirata,P. Ho, J.-P. Kneib, O. L. Fevre, L. Lin, S. More, H. Mu-rayama, T. Nagao, M. Ouchi, M. Seiffert, J. D. Silver-man, L. Sodre, D. N. Spergel, M. A. Strauss, H. Sugai,Y. Suto, H. Takami, and R. Wyse, PASJ 66, R1 (2014),arXiv:1206.0737.

[13] https://www.desi.lbl.gov.[14] https://www.lsst.org.[15] https://www.euclid-ec.org.[16] https://roman.gsfc.nasa.gov.

32

[17] D. Spergel, N. Gehrels, C. Baltay, D. Bennett, J. Breck-inridge, M. Donahue, A. Dressler, B. S. Gaudi,T. Greene, O. Guyon, C. Hirata, J. Kalirai, N. J. Kas-din, B. Macintosh, W. Moos, S. Perlmutter, M. Post-man, B. Rauscher, J. Rhodes, Y. Wang, D. Wein-berg, D. Benford, M. Hudson, W.-S. Jeong, Y. Mellier,W. Traub, T. Yamada, P. Capak, J. Colbert, D. Mas-ters, M. Penny, D. Savransky, D. Stern, N. Zimmerman,R. Barry, L. Bartusek, K. Carpenter, E. Cheng, D. Con-tent, F. Dekens, R. Demers, K. Grady, C. Jackson,G. Kuan, J. Kruk, M. Melton, B. Nemati, B. Parvin,I. Poberezhskiy, C. Peddie, J. Ruffa, J. K. Wallace,A. Whipple, E. Wollack, and F. Zhao, ArXiv e-prints(2015), arXiv:1503.03757 [astro-ph.IM].

[18] M. Vogelsberger, S. Genel, V. Springel, P. Torrey,D. Sijacki, D. Xu, G. Snyder, S. Bird, D. Nelson,and L. Hernquist, Nature (London) 509, 177 (2014),arXiv:1405.1418 [astro-ph.CO].

[19] V. Springel, R. Pakmor, A. Pillepich, R. Wein-berger, D. Nelson, L. Hernquist, M. Vogelsberger,S. Genel, P. Torrey, F. Marinacci, and J. Naiman,Mon. Not. Roy. Astron. Soc. 475, 676 (2018),arXiv:1707.03397 [astro-ph.GA].

[20] U. Seljak, A. Makarov, P. McDonald, S. F. Anderson,N. A. Bahcall, J. Brinkmann, S. Burles, R. Cen, M. Doi,J. E. Gunn, Z. Ivezic, S. Kent, J. Loveday, R. H. Lupton,J. A. Munn, R. C. Nichol, J. P. Ostriker, D. J. Schlegel,D. P. Schneider, M. Tegmark, D. E. Berk, D. H. Wein-berg, and D. G. York, Phys. Rev. D 71, 103515 (2005),arXiv:astro-ph/0407372 [astro-ph].

[21] M. Oguri and M. Takada, Phys. Rev. D 83, 023008(2011), arXiv:1010.0744 [astro-ph.CO].

[22] M. Cacciato, F. C. van den Bosch, S. More, H. Mo, andX. Yang, Mon. Not. Roy. Astron. Soc. 430, 767 (2013),arXiv:1207.0503 [astro-ph.CO].

[23] R. Mandelbaum, A. Slosar, T. Baldauf, U. Seljak,C. M. Hirata, R. Nakajima, R. Reyes, and R. E.Smith, Mon. Not. Roy. Astron. Soc. 432, 1544 (2013),arXiv:1207.1120.

[24] H. Miyatake, S. More, R. Mandelbaum, M. Takada,D. N. Spergel, J.-P. Kneib, D. P. Schneider,J. Brinkmann, and J. R. Brownstein, Astrophys. J.806, 1 (2015), arXiv:1311.1480.

[25] S. More, H. Miyatake, R. Mandelbaum, M. Takada,D. N. Spergel, J. R. Brownstein, and D. P. Schneider,Astrophys. J. 806, 2 (2015), arXiv:1407.1856.

[26] S. Sugiyama, M. Takada, Y. Kobayashi, H. Miyatake,M. Shirasaki, T. Nishimichi, and Y. Park, Phys. Rev.D 102, 083520 (2020), arXiv:2008.06873 [astro-ph.CO].

[27] F. Bernardeau, S. Colombi, E. Gaztanaga, andR. Scoccimarro, Phys. Rep. 367, 1 (2002), arXiv:astro-ph/0112551.

[28] V. Desjacques, D. Jeong, and F. Schmidt, Phys. Rep.733, 1 (2018), arXiv:1611.09787 [astro-ph.CO].

[29] E. Krause, T. F. Eifler, J. Zuntz, O. Friedrich, M. A.Troxel, S. Dodelson, J. Blazek, L. F. Secco, N. Mac-Crann, E. Baxter, C. Chang, N. Chen, M. Crocce,J. DeRose, A. Ferte, N. Kokron, F. Lacasa, V. Miranda,Y. Omori, A. Porredon, R. Rosenfeld, S. Samuroff,M. Wang, R. H. Wechsler, T. M. C. Abbott, F. B.Abdalla, S. Allam, J. Annis, K. Bechtol, A. Benoit-Levy, G. M. Bernstein, D. Brooks, D. L. Burke,D. Capozzi, M. Carrasco Kind, J. Carretero, C. B.D’Andrea, L. N. da Costa, C. Davis, D. L. DePoy,

S. Desai, H. T. Diehl, J. P. Dietrich, A. E. Evrard,B. Flaugher, P. Fosalba, J. Frieman, J. Garcia-Bellido,E. Gaztanaga, T. Giannantonio, D. Gruen, R. A. Gru-endl, J. Gschwend, G. Gutierrez, K. Honscheid, D. J.James, T. Jeltema, K. Kuehn, S. Kuhlmann, O. La-hav, M. Lima, M. A. G. Maia, M. March, J. L. Mar-shall, P. Martini, F. Menanteau, R. Miquel, R. C.Nichol, A. A. Plazas, A. K. Romer, E. S. Rykoff,E. Sanchez, V. Scarpine, R. Schindler, M. Schubnell,I. Sevilla-Noarbe, M. Smith, M. Soares-Santos, F. So-breira, E. Suchyta, M. E. C. Swanson, G. Tarle, D. L.Tucker, V. Vikram, A. R. Walker, and J. Weller, ArXive-prints (2017), arXiv:1706.09359.

[30] N. MacCrann, J. DeRose, R. H. Wechsler, J. Blazek,E. Gaztanaga, M. Crocce, E. S. Rykoff, M. R. Becker,B. Jain, E. Krause, T. F. Eifler, D. Gruen, J. Zuntz,M. A. Troxel, J. Elvin-Poole, J. Prat, M. Wang, S. Do-delson, A. Kravtsov, P. Fosalba, M. T. Busha, A. E.Evrard, D. Huterer, T. M. C. Abbott, F. B. Ab-dalla, S. Allam, J. Annis, S. Avila, G. M. Bernstein,D. Brooks, E. Buckley-Geer, D. L. Burke, A. CarneroRosell, M. Carrasco Kind, J. Carretero, F. J. Cas-tander, R. Cawthon, C. E. Cunha, C. B. D’Andrea,L. N. da Costa, C. Davis, J. De Vicente, H. T.Diehl, P. Doel, J. Frieman, J. Garcıa-Bellido, D. W.Gerdes, R. A. Gruendl, G. Gutierrez, W. G. Hart-ley, D. Hollowood, K. Honscheid, B. Hoyle, D. J.James, T. Jeltema, D. Kirk, K. Kuehn, N. Kuropatkin,M. Lima, M. A. G. Maia, J. L. Marshall, F. Menan-teau, R. Miquel, A. A. Plazas, A. Roodman, E. Sanchez,V. Scarpine, M. Schubnell, I. Sevilla-Noarbe, M. Smith,R. C. Smith, M. Soares-Santos, F. Sobreira, E. Suchyta,M. E. C. Swanson, G. Tarle, D. Thomas, A. R. Walker,J. Weller, and DES Collaboration, Mon. Not. Roy. As-tron. Soc. 480, 4614 (2018), arXiv:1803.09795 [astro-ph.CO].

[31] T. Nishimichi, G. D’Amico, M. M. Ivanov, L. Sena-tore, M. Simonovic, M. Takada, M. Zaldarriaga, andP. Zhang, arXiv e-prints , arXiv:2003.08277 (2020),arXiv:2003.08277 [astro-ph.CO].

[32] P. J. E. Peebles, The large-scale structure of the universe(1980).

[33] U. Seljak, Mon. Not. Roy. Astron. Soc. 318, 203 (2000),arXiv:astro-ph/0001493.

[34] J. A. Peacock and R. E. Smith, Mon. Not. Roy. As-tron. Soc. 318, 1144 (2000), arXiv:astro-ph/0005010.

[35] C. Ma and J. N. Fry, Astrophys. J. 543, 503 (2000),arXiv:astro-ph/0003343.

[36] R. Scoccimarro, R. K. Sheth, L. Hui, and B. Jain, As-trophys. J. 546, 20 (2001), arXiv:astro-ph/0006319.

[37] A. Cooray and R. Sheth, Phys. Rep. 372, 1 (2002),arXiv:astro-ph/0206508.

[38] Y. P. Jing, H. J. Mo, and G. Borner, Astrophys. J. 494,1 (1998), arXiv:astro-ph/9707106 [astro-ph].

[39] Z. Zheng et al., Astrophys. J. 633, 791 (2005),arXiv:astro-ph/0408564.

[40] T. Nishimichi, M. Takada, R. Takahashi, K. Osato,M. Shirasaki, T. Oogi, H. Miyatake, M. Oguri, R. Mu-rata, Y. Kobayashi, and N. Yoshida, Astrophys. J. 884,29 (2019), arXiv:1811.09504 [astro-ph.CO].

[41] S. Alam, M. Ata, S. Bailey, F. Beutler, D. Bizyaev,J. A. Blazek, A. S. Bolton, J. R. Brownstein, A. Burden,C.-H. Chuang, J. Comparat, A. J. Cuesta, K. S. Daw-son, D. J. Eisenstein, S. Escoffier, H. Gil-Marın, J. N.

33

Grieb, N. Hand, S. Ho, K. Kinemuchi, D. Kirkby, F. Ki-taura, E. Malanushenko, V. Malanushenko, C. Maras-ton, C. K. McBride, R. C. Nichol, M. D. Olm-stead, D. Oravetz, N. Padmanabhan, N. Palanque-Delabrouille, K. Pan, M. Pellejero-Ibanez, W. J. Per-cival, P. Petitjean, F. Prada, A. M. Price-Whelan,B. A. Reid, S. A. Rodrıguez-Torres, N. A. Roe, A. J.Ross, N. P. Ross, G. Rossi, J. A. Rubino-Martın,S. Saito, S. Salazar-Albornoz, L. Samushia, A. G.Sanchez, S. Satpathy, D. J. Schlegel, D. P. Schneider,C. G. Scoccola, H.-J. Seo, E. S. Sheldon, A. Simmons,A. Slosar, M. A. Strauss, M. E. C. Swanson, D. Thomas,J. L. Tinker, R. Tojeiro, M. V. Magana, J. A. Vazquez,L. Verde, D. A. Wake, Y. Wang, D. H. Weinberg,M. White, W. M. Wood-Vasey, C. Yeche, I. Zehavi,Z. Zhai, and G.-B. Zhao, Mon. Not. Roy. Astron. Soc.470, 2617 (2017), arXiv:1607.03155 [astro-ph.CO].

[42] R. Mandelbaum, H. Miyatake, T. Hamana, M. Oguri,M. Simet, R. Armstrong, J. Bosch, R. Murata,F. Lanusse, A. Leauthaud, J. Coupon, S. More,M. Takada, S. Miyazaki, J. S. Speagle, M. Shirasaki,C. Sifon, S. Huang, A. J. Nishizawa, E. Medezinski,Y. Okura, N. Okabe, N. Czakon, R. Takahashi, W. R.Coulton, C. Hikage, Y. Komiyama, R. H. Lupton, M. A.Strauss, M. Tanaka, and Y. Utsumi, PASJ 70, S25(2018), arXiv:1705.06745 [astro-ph.CO].

[43] Y. Kobayashi, T. Nishimichi, M. Takada, andR. Takahashi, Phys. Rev. D 101, 023510 (2020),arXiv:1907.08515 [astro-ph.CO].

[44] R. Mandelbaum, U. Seljak, G. Kauffmann, C. M. Hi-rata, and J. Brinkmann, Mon. Not. Roy. Astron. Soc.368, 715 (2006), arXiv:astro-ph/0511164.

[45] A. Schneider and R. Teyssier, JCAP 2015, 049 (2015),arXiv:1510.06034 [astro-ph.CO].

[46] A. Schneider, R. Teyssier, J. Stadel, N. E. Chisari,A. M. C. Le Brun, A. Amara, and A. Refregier, JCAP2019, 020 (2019), arXiv:1810.08629 [astro-ph.CO].

[47] F. C. van den Bosch, S. More, M. Cacciato, H. Mo, andX. Yang, Mon. Not. Roy. Astron. Soc. 430, 725 (2013),arXiv:1206.6890 [astro-ph.CO].

[48] Y. Kobayashi, T. Nishimichi, M. Takada, R. Taka-hashi, and K. Osato, Phys. Rev. D 102, 063504 (2020),arXiv:2005.06122 [astro-ph.CO].

[49] https://jila.colorado.edu/~ajsh/FFTLog/index.

html.[50] J. F. Navarro, C. S. Frenk, and S. D. M. White, Astro-

phys. J. 490, 493 (1997), arXiv:astro-ph/9611107.[51] B. Diemer and A. V. Kravtsov, Astrophys. J. 799, 108

(2015), arXiv:1407.4730 [astro-ph.CO].[52] https://bdiemer.bitbucket.io/colossus/.[53] B. Diemer, Astrophys. J. Suppl. 239, 35 (2018),

arXiv:1712.04512 [astro-ph.CO].[54] C. Hikage, R. Mandelbaum, M. Takada, and D. N.

Spergel, Mon. Not. Roy. Astron. Soc. 435, 2345 (2013),arXiv:1211.1009 [astro-ph.CO].

[55] S. Masaki, C. Hikage, M. Takada, D. N. Spergel, andN. Sugiyama, Mon. Not. Roy. Astron. Soc. 433, 3506(2013), arXiv:1211.7077 [astro-ph.CO].

[56] I. Mohammed and U. Seljak, Mon. Not. Roy. As-tron. Soc. 445, 3382 (2014), arXiv:1407.0060.

[57] O. H. E. Philcox, D. N. Spergel, and F. Villaescusa-Navarro, Phys. Rev. D 101, 123520 (2020),arXiv:2004.09515 [astro-ph.CO].

[58] B. Hadzhiyska, S. Bose, D. Eisenstein, andL. Hernquist, arXiv e-prints , arXiv:2008.04913 (2020),arXiv:2008.04913 [astro-ph.CO].

[59] Due to the line-of-sight projection, the BAO features,which are around 100 h−1Mpc in the three-dimensionalcorrelation function, appear at shorter projected sepa-ration.

[60] S. More, Astrophys. J. Lett. 777, L26 (2013),arXiv:1309.2943 [astro-ph.CO].

[61] M. Tanaka, J. Coupon, B.-C. Hsieh, S. Mineo, A. J.Nishizawa, J. Speagle, H. Furusawa, S. Miyazaki, andH. Murayama, PASJ 70, S9 (2018), arXiv:1704.05988[astro-ph.GA].

[62] G. M. Bernstein and M. Jarvis, Astron. J. 123, 583(2002), astro-ph/0107431.

[63] E. S. Sheldon, D. E. Johnston, J. A. Frieman, R. Scran-ton, T. A. McKay, A. J. Connolly, T. Budavari, I. Ze-havi, N. A. Bahcall, J. Brinkmann, and M. Fukugita,Astron. J. 127, 2544 (2004), arXiv:astro-ph/0312036[astro-ph].

[64] R. Mandelbaum, A. Tasitsiomi, U. Seljak, A. V.Kravtsov, and R. H. Wechsler, Mon. Not. Roy. As-tron. Soc. 362, 1451 (2005), arXiv:astro-ph/0410711.

[65] M. Shirasaki and M. Takada, Mon. Not. Roy. As-tron. Soc. 478, 4277 (2018), arXiv:1802.09696 [astro-ph.CO].

[66] D. Huterer, M. Takada, G. Bernstein, and B. Jain,Mon. Not. Roy. Astron. Soc. 366, 101 (2006),arXiv:astro-ph/0506030.

[67] S. Bridle, S. T. Balan, M. Bethge, M. Gentile,S. Harmeling, C. Heymans, M. Hirsch, R. Hos-seini, M. Jarvis, D. Kirk, T. Kitching, K. Kuijken,A. Lewis, S. Paulin-Henriksson, B. Scholkopf, M. Ve-lander, L. Voigt, D. Witherick, A. Amara, G. Bern-stein, F. Courbin, M. Gill, A. Heavens, R. Man-delbaum, R. Massey, B. Moghaddam, A. Rassat,A. Refregier, J. Rhodes, T. Schrabback, J. Shawe-Taylor, M. Shmakova, L. van Waerbeke, andD. Wittman, Mon. Not. Roy. Astron. Soc. 405, 2044(2010), arXiv:0908.0945 [astro-ph.CO].

[68] N. Kaiser, Mon. Not. Roy. Astron. Soc. 227, 1 (1987).[69] P. S. Behroozi, R. H. Wechsler, and H.-Y. Wu, As-

trophys. J. 762, 109 (2013), arXiv:1110.4372 [astro-ph.CO].

[70] https://www.sdss.org/dr11/.[71] A. Jenkins, C. S. Frenk, F. R. Pearce, P. A. Thomas,

J. M. Colberg, S. D. M. White, H. M. P. Couchman,J. A. Peacock, G. Efstathiou, and A. H. Nelson, Astro-phys. J. 499, 20 (1998), arXiv:astro-ph/9709010 [astro-ph].

[72] P. Valageas and T. Nishimichi, Astronomy & As-trophysics 527, A87 (2011), arXiv:1009.0597 [astro-ph.CO].

[73] R. Takahashi, M. Sato, T. Nishimichi, A. Taruya, andM. Oguri, The Astrophysical Journal 761, 152 (2012).

[74] We use the slightly different number of realizations for∆Σ and wp, because the N -body simulation data for the4 realizations in the difference were lost in the middleof this work. The difference is small compared to thestatistical measurement errors for the SDSS and HSCdata, and the main results of this paper are not changed.

[75] R. Takahashi, T. Hamana, M. Shirasaki, T. Namikawa,T. Nishimichi, K. Osato, and K. Shiroyama, Astrophys.J. 850, 24 (2017), arXiv:1706.01472 [astro-ph.CO].

34

[76] M. Shirasaki, M. Takada, H. Miyatake, R. Taka-hashi, T. Hamana, T. Nishimichi, and R. Mu-rata, Mon. Not. Roy. Astron. Soc. 470, 3476 (2017),arXiv:1607.08679 [astro-ph.CO].

[77] M. Shirasaki, T. Hamana, M. Takada, R. Takahashi,and H. Miyatake, Mon. Not. Roy. Astron. Soc. 486, 52(2019), arXiv:1901.09488 [astro-ph.CO].

[78] D. Kodwani, D. Alonso, and P. Ferreira, The OpenJournal of Astrophysics 2, 3 (2019), arXiv:1811.11584[astro-ph.CO].

[79] Precisely speaking, the binning of R does not have anexact cut at 2, 3 or 30 h−1Mpc for the lower and uppercuts. We employ the 30 bins logarithmically spaced over0.05 ≤ R/[h−1Mpc] ≤ 80, and use the bins residing thefitting range of ∆Σ and wp.

[80] F. Feroz, M. P. Hobson, and M. Bridges,Mon. Not. Roy. Astron. Soc. 398, 1601 (2009),arXiv:0809.3437 [astro-ph].

[81] B. Audren, J. Lesgourgues, K. Benabed, andS. Prunet, JCAP 2013, 001 (2013), arXiv:1210.7183[astro-ph.CO].

[82] B. Joachimi, C. A. Lin, M. Asgari, T. Troster, C. Hey-mans, H. Hildebrandt, F. Kohlinger, A. G. Sanchez,A. H. Wright, M. Bilicki, C. Blake, J. L. van denBusch, M. Crocce, A. Dvornik, T. Erben, F. Get-man, B. Giblin, H. Hoekstra, A. Kannawadi, K. Kui-jken, N. R. Napolitano, P. Schneider, R. Scoccimarro,E. Sellentin, H. Y. Shan, M. von Wietersheim-Kramsta,and J. Zuntz, arXiv e-prints , arXiv:2007.01844 (2020),arXiv:2007.01844 [astro-ph.CO].

[83] L. Gao, V. Springel, and S. D. M. White,Mon. Not. Roy. Astron. Soc. 363, L66 (2005),arXiv:astro-ph/0506510 [astro-ph].

[84] R. H. Wechsler, A. R. Zentner, J. S. Bullock, A. V.Kravtsov, and B. Allgood, Astrophys. J. 652, 71 (2006),arXiv:astro-ph/0512416 [astro-ph].

[85] N. Dalal, M. White, J. R. Bond, and A. Shirokov, As-trophys. J. 687, 12 (2008), arXiv:0803.3453 [astro-ph].

[86] A. Leauthaud, S. Saito, S. Hilbert, A. Barreira, S. More,M. White, S. Alam, P. Behroozi, K. Bundy, J. Coupon,T. Erben, C. Heymans, H. Hildebrandt, R. Mandel-baum, L. Miller, B. Moraes, M. E. S. Pereira, S. A.Rodrıguez-Torres, F. Schmidt, H.-Y. Shan, M. Viel,and F. Villaescusa-Navarro, Mon. Not. Roy. As-tron. Soc. 467, 3024 (2017), arXiv:1611.08606 [astro-ph.CO].

[87] E. Semboloni, H. Hoekstra, and J. Schaye,Mon. Not. Roy. Astron. Soc. 434, 148 (2013),arXiv:1210.7303 [astro-ph.CO].

[88] The code of analytical halo model is available fromhttps://github.com/surhudm/aum.

[89] J. L. Tinker, D. H. Weinberg, Z. Zheng, and I. Zehavi,Astrophys. J. 631, 41 (2005), arXiv:astro-ph/0411777[astro-ph].

[90] J. L. Tinker, B. E. Robertson, A. V. Kravtsov,A. Klypin, M. S. Warren, G. Yepes, and S. Gottlober,Astrophys. J. 724, 878 (2010), arXiv:1001.3162 [astro-ph.CO].

[91] Y. Li, W. Hu, and M. Takada, Phys. Rev. D 93, 063507(2016), arXiv:1511.01454.

[92] T. Baldauf, U. Seljak, L. Senatore, and M. Zaldarriaga,JCAP 9, 007 (2016), arXiv:1511.01465.

[93] T. Lazeyras, C. Wagner, T. Baldauf, and F. Schmidt,JCAP 2, 018 (2016), arXiv:1511.01096.

[94] M. Oguri, Y.-T. Lin, S.-C. Lin, A. J. Nishizawa,A. More, S. More, B.-C. Hsieh, E. Medezinski, H. Miy-atake, H.-Y. Jian, L. Lin, M. Takada, N. Okabe,J. S. Speagle, J. Coupon, A. Leauthaud, R. H. Lup-ton, S. Miyazaki, P. A. Price, M. Tanaka, I. N. Chiu,Y. Komiyama, Y. Okura, M. M. Tanaka, and T. Usuda,PASJ 70, S20 (2018), arXiv:1701.00818 [astro-ph.CO].

[95] H. Miyatake, N. Battaglia, M. Hilton, E. Medezin-ski, A. J. Nishizawa, S. More, S. Aiola, N. Bahcall,J. R. Bond, E. Calabrese, S. K. Choi, M. J. Devlin,J. Dunkley, R. Dunner, B. Fuzia, P. Gallardo, M. Gralla,M. Hasselfield, M. Halpern, C. Hikage, J. C. Hill, A. D.Hincks, R. Hlozek, K. Huffenberger, J. P. Hughes,B. Koopman, A. Kosowsky, T. Louis, M. S. Mad-havacheril, J. McMahon, R. Mandelbaum, T. A. Mar-riage, L. Maurin, S. Miyazaki, K. Moodley, R. Murata,S. Naess, L. Newburgh, M. D. Niemack, T. Nishimichi,N. Okabe, M. Oguri, K. Osato, L. Page, B. Partridge,N. Robertson, N. Sehgal, B. Sherwin, M. Shirasaki,J. Sievers, C. Sifon, S. Simon, D. N. Spergel, S. T.Staggs, G. Stein, M. Takada, H. Trac, K. Umetsu,A. van Engelen, and E. J. Wollack, Astrophys. J. 875,63 (2019), arXiv:1804.05873 [astro-ph.CO].

[96] R. Mandelbaum, F. Lanusse, A. Leauthaud, R. Arm-strong, M. Simet, H. Miyatake, J. E. Meyers,J. Bosch, R. Murata, S. Miyazaki, and M. Tanaka,Mon. Not. Roy. Astron. Soc. 481, 3170 (2018),arXiv:1710.00885 [astro-ph.CO].

[97] J. Hartlap, P. Simon, and P. Schneider, Astronomy &Astrophysics 464, 399 (2007), arXiv:astro-ph/0608064[astro-ph].

[98] A. Taylor, B. Joachimi, and T. Kitching,Mon. Not. Roy. Astron. Soc. 432, 1928 (2013),arXiv:1212.4359 [astro-ph.CO].

[99] S. Dodelson and M. D. Schneider, Phys. Rev. D 88,063537 (2013), arXiv:1304.2593 [astro-ph.CO].

[100] D. J. Paz and A. G. Sanchez, Mon. Not. Roy. As-tron. Soc. 454, 4326 (2015), arXiv:1508.03162 [astro-ph.CO].

[101] O. Friedrich and T. Eifler, Mon. Not. Roy. Astron. Soc.473, 4150 (2018), arXiv:1703.07786 [astro-ph.IM].

[102] http://cosmo.phys.hirosaki-u.ac.jp/takahasi/

allsky_raytracing/.[103] G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel,

C. L. Bennett, J. Dunkley, M. R. Nolta, M. Halpern,R. S. Hill, N. Odegard, L. Page, K. M. Smith, J. L.Weiland, B. Gold, N. Jarosik, A. Kogut, M. Limon, S. S.Meyer, G. S. Tucker, E. Wollack, and E. L. Wright,ApJS 208, 19 (2013).

[104] T. Hamana and Y. Mellier, Mon. Not. Roy. Astron. Soc.327, 169 (2001), astro-ph/0101333.

[105] M. R. Becker, MNRAS 435, 115 (2013).[106] M. Shirasaki, T. Hamana, and N. Yoshida,

Mon. Not. Roy. Astron. Soc. 453, 3043 (2015),arXiv:1504.05672.

[107] P. Fosalba, E. Gaztanaga, F. J. Castander, andM. Manera, Mon. Not. Roy. Astron. Soc. 391, 435(2008), arXiv:0711.1540 [astro-ph].

[108] K. M. Gorski, E. Hivon, A. J. Banday, B. D. Wandelt,F. K. Hansen, M. Reinecke, and M. Bartelmann, ApJ622, 759 (2005), astro-ph/0409513.

[109] R. Mandelbaum, H. Miyatake, T. Hamana, M. Oguri,M. Simet, R. Armstrong, J. Bosch, R. Murata,F. Lanusse, A. Leauthaud, J. Coupon, S. More,

35

M. Takada, S. Miyazaki, J. S. Speagle, M. Shirasaki,C. Sifon, S. Huang, A. J. Nishizawa, E. Medezinski,Y. Okura, N. Okabe, N. Czakon, R. Takahashi, W. R.Coulton, C. Hikage, Y. Komiyama, R. H. Lupton, M. A.Strauss, M. Tanaka, and Y. Utsumi, PASJ 70, S25(2018), arXiv:1705.06745 [astro-ph.CO].

[110] M. Shirasaki and N. Yoshida, Astrophys. J. 786, 43(2014), arXiv:1312.5032 [astro-ph.CO].

[111] http://gfarm.ipmu.jp/~surhud/.[112] M. Takada and W. Hu, Phys. Rev. D 87, 123504 (2013),

arXiv:1302.6994 [astro-ph.CO].[113] Y. Li, W. Hu, and M. Takada, Phys. Rev. D 89, 083519

(2014), arXiv:1401.0385 [astro-ph.CO].[114] R. Takahashi, T. Nishimichi, M. Takada, M. Shirasaki,

and K. Shiroyama, Mon. Not. Roy. Astron. Soc. 482,4253 (2019), arXiv:1805.11629 [astro-ph.CO].

Appendix A: Cosmological dependence of the 2-haloterm of galaxy correlation functions

The cosmological information in the joint probes cos-mology using ∆Σ and wp lies in the 2-halo terms of ∆Σand wp. In particular, an advantage in the use of DarkEmulator is that it includes all the nonlinear effectssuch as the nonlinear matter clustering, the nonlinearhalo bias and the halo exclusion effect that are very dif-ficult to accurately calibrate unless N -body simulationsare employed as done in Dark Emulator. In this ap-pendix, we study how these nonlinear effects cause scale-dependent variations in the 2-halo term by changes inthe cosmological parameters.

Fig. 19 shows how a change in σ8 or Ωm0 causes ascale-dependent change in the 2-halo term of the three-dimensional correlation function, ξgg,2h, while other pa-rameters including the HOD parameters are kept to theirfiducial values. Note that we use Dark Emulator tocompute these results, and we here focus on the galaxyauto-correlation function because most of the cosmologi-cal information for the SDSS and HSC datasets are fromwp that is obtained from the projection of ξgg. First,for comparison, the upper panel shows the change in thematter correlation function, ξmm, relative to that for thefiducial model (Planck cosmology). The figure shows thata change in σ8 causes a fairly sale-independent change inξmm at scales r . 70 h−1Mpc which we consider through-out this paper. On the other hand, a change in Ωm0

causes a scale-dependent change in ξmm.

The middle panel shows the fractional change inξgg,2h/ξmm relative to the ratio for the fiducial model.Note that we here consider the fiducial model of HOD forLOWZ-like galaxies at z = 0.25. The reason we considerthis ratio is that, if ξgg follows the linear theory predic-tion as given by ξgg = b2ξmm with a scale-independentcoefficient b, the change becomes scale-independent forthe change in the cosmological parameters. Here, for thesake of clarity, we consider only the 2-halo term of ξgg us-ing Eq. (16). The figure shows that a change in σ8 causesa scale-dependent change in the ratio, which should arises

0.8

0.9

1.0

1.1

1.2

ξ mm/ξ

fid mm

0.8

0.9

1.0

1.1

1.2

(ξgg,2

h/ξ

mm

)/(ξ

fid gg,2

h/ξ

fid mm

)

100 101 102

r [h−1Mpc]

0.8

0.9

1.0

1.1

1.2

(ξgg,c

c/ξ m

m)/

(ξfid gg,c

c/ξfi

dm

m)

Ωm

σ28

5%

10%

FIG. 19. Fractional changes in the 2-halo term of the three-dimensional galaxy-galaxy correlation function relative tothat for the fiducial model, when varying either of σ8 or Ωm0.Note that other model parameters keep fixed to their fiducialvalues. The middle panel shows the fractional change for theratio ξgg,2h/ξmm, because the ratio becomes scale-independentif the linear theory holds, which predicts ξgg = b2ξmm with ascale-independent coefficient b. The lower panel shows the re-sult for the two-point correlation function of central galaxiesthat includes only the two-halo term by definition. The dif-ferent lines show the results when ±5% and ±10% changes inσ2

8 and Ωm0, and the solid or dashed lines correspond to theresults when increasing or decreasing the parameter by theamount from the fiducial value. Here we consider the HODfor LOWZ-like galaxies at z = 0.25.

from the scale-dependent change in the halo bias and thehalo exclusion effect by the change in σ8. The change inΩm0 is also found to cause a scale-dependent change inthe ratio, again via the dependence of halo bias on Ωm0.Thus Dark Emulator includes these complex depen-dences of ξgg on the cosmological parameters. For further

36

comparison, the lower panel shows the results for ξgg,cc,i.e the two-point correlation function of central galaxies,which includes only the 2-halo term by definition. Theresults appear to be similar to those in the middle panel.

Thus Dark Emulator includes these complex depen-dences of the galaxy-galaxy correlation function on thecosmological parameters (σ8 and Ωm0). These depen-dences are difficult to accurately calibrate analyticallye.g. by the perturbation theory, and the accurate cali-bration requires the use of N -body simulations as donein the development of Dark Emulator. These complexcosmological dependences of ξgg are an advantage of ourmethod.

Appendix B: Covariance estimation based on mockcatalogs

In this appendix, we describe the estimation of statisti-cal uncertainties in the galaxy-galaxy weak lensing profile(∆Σ) by using a set of synthetic observational datasets,i.e. the covariance matrix.

A robust estimation of the covariance matrix is oneof the most important subjects in modern cosmology inpractice [e.g., see Refs. 97–101]. In this paper, we use aset of numerical simulations in Ref. [75] to construct real-istic mock catalogs of galaxy shapes as well as the tracersof large-scale structures. We then adopt the same analy-sis pipeline to measure ∆Σ from the mock catalogs as wedo for actual measurements. Our mock measurementsof the lensing signals have 2268 realizations in total, al-lowing us to evaluate the covariance matrix of ∆Σ forthe SDSS-like galaxies at multiple redshifts in a rigorousway. In the following, we describe how to produce mockcatalogs from the simulations incorporated with observa-tional data and show the validation of our mock lensinganalyses.

1. Massive production of mock catalogs

Full-sky simulations of lensing and halos

We first briefly introduce the full-sky ray-tracing sim-ulations and the halo catalogs in the line-cone simulationrealization, developed in Ref. [75] (The full-sky simula-tion data are freely available from [102]). The full-skysimulations are based on a set of N -body simulationswith 20483 particles in cosmological volumes. Ref. [75]adopted the standard ΛCDM cosmology with the follow-ing cosmological parameters: the CDM density param-eter Ωcdm = 0.233, the baryon density Ωb = 0.046, thematter density Ωm = Ωcdm + Ωb = 0.279, the cosmologi-cal constant ΩΛ = 0.721, the Hubble parameter h = 0.7,the amplitude of density fluctuations σ8 = 0.82, and thespectral index ns = 0.97. Note that the cosmologicalmodel in the simulation is consistent with the WMAP 9year cosmology [103].

Full-sky weak gravitational lensing simulations havebeen performed with the standard multiple lens-plane al-gorithm [104–106]. In this simulation, one can take intoaccount the light-ray deflection on the celestial sphereby using the projected matter density field given in theformat of spherical shell (see the similar approach forRef. [107]). The simulations used the projected matterfields in 38 shells in total, each of which was computedby projecting N -body simulation realization over a radialwidth of 150h−1Mpc, in order to make the light-conesimulation covering a cosmological volume up to z = 5.3.As a result, the lensing simulations consist of the shearfield at each of 38 different source redshifts with angularresolution of 0.43 arcmin. Each simulation data is givenin the HEALPix format [108]. The radial depth be-tween nearest source redshifts is set to be 150h−1Mpc incomoving distance, corresponding to the redshift intervalof 0.05− 0.1 for z <∼ 1.

In each output of the N -body simulation, Ref. [75]identified dark matter halos using the Rockstar algo-rithm [69]. In the following, we define the halo massby using the spherical overdensity criterion: M200m =200ρm0(4π/3)R3

200m. Individual halos in N -body boxesare assigned to the pixels in the celestial sphere with theHEALPix software. It should be noted that the N -bodysimulations allow us to resolve dark matter halos withmasses greater than a few times 1012 h−1M with morethan 50 N -body particles at redshifts z < 0.7, which isa typical redshift range of massive galaxies in the SDSSBOSS survey.

Shapes of background galaxies

For shapes of background galaxies, we use the mockcatalogs produced in Ref. [77]. The mocks are specificto cosmological analyses with the Subaru HSC-Y1 shapecatalog [109] by taking into account various observationaleffects as the survey footprints, inhomogeneous angulardistribution of source galaxies, statistical uncertainties inphotometric redshift estimate, variations in the lensingweight, and the statistical noise in galaxy shape mea-surements including both intrinsic shapes and the mea-surement errors. We produced 2268 mock catalogs from108 full-sky ray-tracing simulations of gravitational lens-ing in Ref. [75] by using a similar approach developed inRefs. [76, 110]. We properly incorporated with the sim-ulated lensing shear and the observed galaxy shape onan object-by-object basis, enabling the mock to share ex-actly the same information of angular positions, redshifts,the lensing weights and the shear responsivity with thereal catalog. The further details are found in Ref. [77].The mock shape catalogs are publicly available at [111].

37

Mock catalogs of lensing galaxies, SDSS-like galaxies

For the foreground galaxies to be used in the galaxy-galaxy lensing analysis, we produce the mock galaxycatalogs assuming the HOD method as summarized inSection III B. As in the mock galaxy shape catalogs,we first extract 2268 realizations of the HSC-Y1 sur-vey windows from 108 full-sky halo catalogs in Ref. [75].We then populate galaxies into halos using the fiducialHOD method whose parameters are chosen to mimicLOWZ and CMASS galaxies in the redshift range of0.15 < z < 0.35, 0.43 < z < 0.55, and 0.55 < z < 0.70,but spanning the entire SDSS BOSS footprint. The HODmethod is used to populate mock LOWZ and CMASSgalaxies in halos of each of the light-cone simulation re-alization.

Besides, we include the redshift-space distortion effectsin mock LOWZ and CMASS galaxies. Along the line ofsight of each host halo, we set the radial velocity of thecentral galaxy to be the same as one of its host halo,while we assign the random velocities to the satellites byfollowing a Gaussian distribution with width given by thevirial velocity dispersion.

2. Validation of our covariance estimation

In this section, we validate the performance of ourcovariance estimation with the mock measurements ofgalaxy-galaxy lensing signals. For comparison, we pre-dict the Gaussian covariances of the lensing signals bya halo-model framework developed in Ref. [65]. In thisframework, we can properly take into account the weightfunction related to the conversion between the lensingshear and the excess surface mass density in the covari-ance matrix. It would be worth noting that the covari-ance model has been extensively validated with a set ofnumerical simulations for a variety of the weight func-tions in the lensing analysis (see Ref. [65] for details).

Fig. 20 shows the comparison of the variance of thelensing signal in each of three redshift bins with the sim-ulation results and their Gaussian predictions. In eachpanel, the simulation results are shown in the coloredpoints, while the solid and dashed lines represent theGaussian covariances with and without the shape noise,respectively. The figure clearly shows that the samplevariance caused by the line-of-sight large scale structuresdominates the mock variance, and the shot noise is dom-inant at the length scale less than 10h−1Mpc.

We then study the off-diagonal elements of the mockcovariance matrices. The left panels in Fig. 21 summa-rize the comparison of the cross correlation coefficient inthe covariance for the lensing signals at the lens redshift0.43 < z < 0.55. This figure shows that our simple Gaus-sian prediction provides a reasonable agreement with theoff-diagonal elements of the mock covariance, indicatingthe super-sample covariance (SSC) [112] is not impor-tant to our galaxy-galaxy lensing analyses. The SSC is

FIG. 20. Comaprison of the diagonal components in the co-variance matrices for the galaxy-galaxy lensing signals. Fromtop to bottom, we show the variances of the lensing signals atthree different redshift bins (the lower panel corresponds tothe case at the higher redshift). The points represent the sim-ulation results based on 2268 realizations, while the solid linesstand for the Gaussian predictions developed in Ref. [65]. Ineach panel, the dashed line shows the variance in the absenceof shape noises, highlighting the contribution from uncorre-lated large-scale structures along a line of the sight. Note thatwe define ∆Σ in units of hM pc−2 in this figure. Hence, theunit in each vertical axis is given by Mpc (M pc−2).

expected to arise from the four-point correlation amongsuper- and sub-survey modes [112, 113], and the recentsimulations have shown that the SSC in the halo-mattercross correlation becomes important only at 1h−1Mpc[114]. At the scale of R ∼ 1h−1Mpc, the statistical un-certainties in our lensing analyses are mostly determinedby the shot noise terms.

The right panel of Fig. 21 shows the cross correlationcoefficient of the mock covariance matrix across threeredshift bins. The lower triangular panel shows the sim-ulation results, while the upper panels represent the dif-ference between the simulation results and the Gaussianpredictions. We find the large cross correlation coeffi-cients in the radius range of R > 10h−1Mpc with a levelof ∼ 0.5 at most for single redshifts and the cross covari-ance between two different redshifts is less prominent.Our Gaussian covariance is in good agreement with thesimulation results, allowing to explain the mock covari-

38

LOWZ CMASS1 CMASS2

R[h−

1M

pc]

10.0

71.3HSCS16A GGL covariance

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

FIG. 21. Comparisons of the off-diagonal elements in the covariance matrices with the simulation results and their Gaussianpredictions. The left panels show the cross correlation coefficient in the covariance at the medium redshift bin (in the redshiftrange of 0.43 < z < 0.55) as a function of radii. In the left, the points show the simulation results, while the lines are forthe Gaussian predictions. Each small panel in the left represents the scale dependence of the cross correlation coefficient.In the right, we show the cross correlation coefficients in the full covariance across three redshift bins. The lower triangularpanels show the simulation results, while the upper one displays the difference between the simulation results and the Gaussianpredictions. In the right, we use the labels of “LOWZ”, ”CMASS1” and “CMASS2” from the lowest to the highest redshiftbins. Note that we limit the length scales to be in the range of R > 10h−1Mpc in the right panels to focus on the samplecovariance. The diagonal elements in the right is set to 1 by construction.

ance with a 20%-level accuracy.

Appendix C: Posterior distribution of parameters ina full-dimension parameter space

For comprehensiveness of our discussion, in Fig. 22 weshow the posterior distribution for all the parameters in-

cluding all the HOD parameters for the CMASS1-likegalaxies. The results for the LOWZ- and CMASS2-likegalaxies at different redshifts are similar to this plot.Even if we use the joint information of wp and ∆Σ, eachof the HOD parameters is not well constrained, but thecosmological parameters are recovered.

39

0.2 0.4

Ωm0

1

2

κ(z

CM

AS

S1)

1

2

α(z

CM

AS

S1)

13

14

15

logM

1(z

CM

AS

S1)

0.2

0.4

0.6

0.8

σ2 lo

gM

(zC

MA

SS

1)

13

14

logM

min

(zC

MA

SS

1)

0.6

0.8

1.0

1.2

S8

0.6

0.8

1.0

1.2

σ8

0.6 0.8 1.0

σ8

0.6 0.8 1.0 1.2

S8

13 14

logMmin(zCMASS1)

0.2 0.5 0.8

σ2logM(zCMASS1)

13 14 15

logM1(zCMASS1)

1 2

α(zCMASS1)

1 2

κ(zCMASS1)

lensing alone

clustering alone

baseline

FIG. 22. Posterior distributions of cosmological parameters and HOD parameters, as in Fig. 9. Here we show the results forall the HOD parameters for the CMASS1-like galaxies (see Table II).