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YITP-19-103Draft version November 20, 2019Preprint typeset using LATEX style emulateapj v. 12/16/11

BIHALOFIT: A NEW FITTING FORMULA OF NON-LINEAR MATTER BISPECTRUM

Ryuichi Takahashi1, Takahiro Nishimichi2,6, Toshiya Namikawa3, Atsushi Taruya2,6, Issha Kayo4, Ken Osato5,Yosuke Kobayashi6, Masato Shirasaki7

1Faculty of Science and Technology, Hirosaki University, 3 Bunkyo-cho, Hirosaki, Aomori 036-8588, Japan2Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan3Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK,

4Department of Liberal Arts, Tokyo University of Technology, Ota-ku, Tokyo 144-8650, Japan,5Institut d’Astrophysique de Paris, Sorbonne Universite, CNRS, UMR 7095, 98bis boulevard Arago, 75014 Paris, France,

6Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study(UTIAS),

The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba, 277-8583, Japan,7National Astronomical Observatory of Japan, Mitaka, Tokyo, 181-8588, Japan

Draft version November 20, 2019

ABSTRACT

We provide an accurate fitting formula of the matter bispectrum in the non-linear regime calibratedby high-resolution cosmological N -body simulations for 41 wCDM models around the Planck 2015best-fit parameters. Our fitting function assumes a similar parameterization as in Halofit for thenon-linear matter power spectrum and so is named BiHalofit. The simulation volume is large enough(> 10Gpc3) to cover almost all measurable triangle configurations of bispectrum in the universe.Our formula can reproduce the matter bispectrum within 10 (15)% accuracy for wavenumber k <3 (10)hMpc−1 at redshifts z = 0–3 for the Planck 2015 model. We also provide a fitting formulato correct baryonic effects such as radiative cooling and active galactic nuclei feedback based on thelatest hydrodynamical simulation, IllustrisTNG. Further, we show that our new formula provides moreaccurate predictions for the weak lensing bispectrum than the existing fitting formulas. The formulawould be quite useful for current and future weak lensing surveys and cosmic microwave backgroundlensing experiments.Subject headings: gravitational lensing: weak – methods: numerical – cosmology: theory – large-scale

structure of universe

1. INTRODUCTION

Observations of cosmic microwave background (CMB)have revealed that the primordial density fluctuations arewell described by a Gaussian field (Planck Collaboration2019). The statistical property of a Gaussian field is fullydescribed by the two-point correlation function, or itsFourier transform, the power spectrum (PS). However, atlate times, the density fluctuations become non-Gaussianthrough gravitational evolution at small scales. Then,we need higher-order statistics to fully characterize thestatistical property of a non-Gaussian field and to accesscosmological information beyond the two-point statistics.The bispectrum (BS), the Fourier transform of the three-point correlation function, is the leading correction to thecommonly used PS.Weak lensing can map a projected density field

via coherent distortion of background galaxies (e.g.,Bartelmann & Schneider 2001). Several weak lensingsurveys are currently in operation, e.g., the SubaruHyper Suprime-Cam (HSC)1, the Dark Energy Survey(DES)2, and the Kilo-Degree Survey (KiDS)3. They pro-vided strong constraints on the cosmological parameterssuch as the matter density Ωm and the amplitude ofdensity fluctuations σ8 from the cosmic-shear two-point

1 https://hsc.mtk.nao.ac.jp/ssp/2 https://www.darkenergysurvey.org3 http://kids.strw.leidenuniv.nl

function (e.g., Abbott et al. 2018; van Uitert et al. 2018;Hamana et al. 2019; Hikage et al. 2019). In the 2020s,ground- and space-based missions will start their opera-tions such as Large Synoptic Survey Telescope (LSST)4,Wide Field Infrared Survey Telescope (WFIRST)5, andEuclid6.The weak lensing BS contains additional and com-

plementary information to the PS. The BS is sensi-tive to smaller-scale and lower-redshift structures thanthe PS because the BS arises from the non-Gaussianity.A joint analysis of both the spectra breaks parame-ter degeneracy and thus provides tighter constraints(e.g., Takada & Jain 2004; Kilbinger & Schneider 2005;Sefusatti et al. 2006; Munshi et al. 2011; Kayo & Takada2013; Byun et al. 2017; Gatti et al. 2019). The BS can becomparable to or more powerful than the PS (Berge et al.2010; Sato & Nishimichi 2013; Coulton et al. 2019). Sev-eral groups previously measured the three-point cosmic-shear statistics from real data and provided use-ful constraints (Bernardeau et al. 2002b; Jarvis et al.2004; Semboloni et al. 2011b; Van Waerbeke et al. 2013;Fu et al. 2014; Simon et al. 2015).CMB lensing is also a promising cosmological probe

to measure the density fluctuations at higher redshift

4 https://www.lsst.org/5 https://wfirst.gsfc.nasa.gov/6 https://www.euclid-ec.org/

http://arxiv.org/abs/1911.07886v1

https://hsc.mtk.nao.ac.jp/ssp/

https://www.darkenergysurvey.org

http://kids.strw.leidenuniv.nl

https://www.lsst.org/

https://wfirst.gsfc.nasa.gov/

https://www.euclid-ec.org/

2 Takahashi et al.

(z ≃ 1–3) than cosmic shear (e.g., Lewis & Challinor2006). Recent CMB experiments have measured lens-ing signals from temperature and polarization fluctua-tions (Planck Collaboration 2018b). The CMB lensing-potential PS provides a rich cosmological informationwhich is complementary to that obtained from the galaxyweak lensing (e.g., Planck Collaboration 2018b). The BSand higher-order spectra induced by the non-Gaussiandensity fluctuations would be important in the futureCMB lensing observations. The non-Gaussianity slightlyaffects the lensing PS (Pratten & Lewis 2016) as well asthe CMB temperature and polarization power spectra(Lewis & Pratten 2016; Marozzi et al. 2018). It also con-taminates the CMB lensing reconstruction (Bohm et al.2016; Beck et al. 2018; Fabbian et al. 2019). On theother hand, future CMB experiments will measure thelensing BS as a useful signal (Namikawa 2016).In this regard, an accurate model of non-linear BS

is highly demanded. Regarding the PS, we shouldprepare a non-linear model with a few percent accu-racy up to k = 10 hMpc−1 to meet the statisticalaccuracy requirements of the forthcoming weak-lensingsurveys7 (Huterer & Takada 2005; Hearin et al. 2012).Though Scoccimarro & Couchman (2001) provided a fit-ting formula of BS calibrated by N -body simulationsand Gil-Marın et al. (2012) improved the formula fur-ther, these overestimate the squeezed BS by a factor 2 inthe worst cases, compared to latest numerical simulations(Fu et al. 2014; Coulton et al. 2019; Namikawa et al.2019; Munshi et al. 2019). In this paper, we constructan improved fitting formula of the non-linear matter BScalibrated by high-resolution cosmological N -body sim-ulations for 41 wCDM (cold dark matter and dark en-ergy with a constant equation of state w) models up tok = 30 hMpc−1 in the range of z = 0–10. We bin bothsimulation data and theoretical prediction in terms ofwavenumbers (k1, k2, k3), which is critically importantfor the accurate calibration. Our formula also takes intoaccount the baryonic effects by using the public hydrody-namic simulation, the IllustrisTNG suite (Nelson et al.2019).

2. THEORY

2.1. Basics

The cosmological density contrast is usually describedby its Fourier transform, δ(k). The matter PS and BSare defined as

P (k1) δD(k1 + k2) = 〈δ(k1)δ(k2)〉,

B(k1, k2, k3) δD(k1 + k2 + k3) = 〈δ(k1)δ(k2)δ(k3)〉,(1)

where δD is the Dirac delta function. Here and through-out this paper, we do not explicitly write the redshiftdependence in the arguments of the functions, but thefollowing discussion is applicable at an arbitrary redshift.At the tree level (i.e., the leading order in perturbation

theory), the matter BS is given by a product of the linear

7 To our best knowledge, there is no study about how accurateBS model is necessary for the current and forthcoming surveys.

matter PS, PL(k), as (e.g., Bernardeau et al. 2002a)

Btree(k1, k2, k3) = 2F2(k1,k2)PL(k1)PL(k2) + 2 perm.(2)

Here, the last term means two permutations: (k1,k2) →(k2,k3) and (k3,k1) applied to the wavevectors in thefirst term. The F2 kernel is

F2(k1,k2) =5

7+

1

2

(

k1k2

+k2k1

)

µ12 +2

7µ212, (3)

where µ12 is the cosine of the angle between k1 and k2,i.e., µ12 = k1 · k2/(k1k2).To explore the non-linear regime, beyond the tree

level, one usually relies on higher-order perturbationtheories (e.g., Scoccimarro et al. 1998; Rampf & Wong2012; Angulo et al. 2015; Hashimoto et al. 2017;Bose & Taruya 2018; Lazanu & Liguori 2018). However,these can be reliable up to the mildly non-linear regime(k . 0.2 hMpc−1). Another strategy is employing thehalo model (e.g., Cooray & Sheth 2002) which is ananalytical model assuming that all the matter is confinedto halos. This model can be used for a wide range ofscale and redshift, but the current accuracy is about30% level (e.g., Lazanu et al. 2016; Bose et al. 2019).The last one is a fitting function calibrated by N -bodysimulations for various scales, epochs and cosmologicalmodels. This is our approach in this paper.

2.2. Previous fitting formulae

Scoccimarro & Couchman (2001) (SC01) provided afitting formula for the non-linear BS. Their functionis similar to the tree level formula (Eq. 2), but re-places the linear PS with a non-linear model and modi-fies the F2 kernel to enhance the amplitude of the BSat small scale. In low-k limit, their formula is con-sistent with the tree level. In high-k limit, the BS isproportional to P (k1)P (k2) + P (k2)P (k3) + P (k3)P (k1)according to the hyper-extended perturbation theory(Scoccimarro & Frieman 1999). Their modified F2 ker-nel contains 6 free parameters, which are fitted by theirN -body result in a range of k < 3 hMpc−1 at 0 ≤ z ≤ 1for four CDM models. Later, Gil-Marın et al. (2012)(GM12) increased the number of free parameters in F2

from 6 to 9 and re-calibrated them from their N -bodysimulations in a relatively narrow range of wavenumbers,k < 0.4 hMpc−1, at 0 ≤ z ≤ 1.5 for a single ΛCDMmodel.However, these formulae have several shortcomings

as follows. (i) A non-linear PS model must be sup-plied to their formula as a basic building block, suchas Halofit (Smith et al. 2003; Takahashi et al. 2012), thehalo model (Mead et al. 2015), or Cosmic Emulator(Lawrence et al. 2017). Therefore, the user should pre-pare the PS fitting formula as well. Furthermore, thesePS models still exhibit small but non-negligible discrep-ancies to each other, typically a few percent level (e.g.,Schneider et al. 2016), which degrades the BS accuracy.(ii) As indicated by Namikawa et al. (2019), these modelsoverestimate the squeezed BS (i.e., for the configurationof k1 ≃ k2 ≫ k3). (iii) Their fitting range of k and z isnarrow. The current weak-lensing surveys measure thecorrelation function down to arcmin scale, which meansthat we need a calibration up to k = 10 hMpc−1. In ad-

Accurate fitting formula of matter bispectrum 3

−1.2

−1

−0.8σ 8

w

Ωm

0.4

0.6

0.8

1

1.2

0.2 0.3 0.4

σ 8w

Ωm

Fig. 1.— Distribution of cosmological parameters in the 41 mod-els. The central red diamond is the Planck 2015 best-fit ΛCDM,while the others are 40 wCDM models. The latter are divided intothree groups in terms of S8 ≡ σ8(Ωm/0.3)0.5; the magenta (cyan)dots have S8 > 1.0 (< 0.6), and the grays are the rest. The dashedcurves are the boundaries (S8 = 0.6 and 1.0).

dition, the CMB lensing probes high-redshift structures(z ≃ 1–3), requiring a calibration up to z = 10. (iv)They did not include the baryonic effects which are im-portant at k & 1 hMpc−1.

2.3. Our fitting formula

We construct a fitting formula based on the halo modelin similar to Halofit for the non-linear PS (Smith et al.2003). The halo model has been frequently used toevaluate the multi-point statistics of non-linear densityfields (see also Appendix A for detailed discussion aboutthe halo-model BS). This model assumes that all parti-cles are contained in halos and decomposes BS to threeterms: one-, two- and tree-halo terms (hereafter denotedas 1h, 2h and 3h terms, respectively). The 1h term de-scribes the correlation in an individual halo, and the2h (3h) term accounts for the correlation among two(three) different halos. The 1h and 3h terms dominate atsmall and large scales, respectively. On the other hand,the 2h term is usually subdominant in most of the tri-angle configurations (except for the squeezed case, seee.g., Valageas & Nishimichi 2011; Valageas et al. 2012).Therefore, we drop this term, but instead enhance the3h term at intermediate scales to absorb the missed 2hcontribution.The fitting function consists of two terms:

B(k1, k2, k3) = B1h(k1, k2, k3) +B3h(k1, k2, k3), (4)

which approaches the tree level formula in low-k limit.The function contains 52 free parameters in total whichwill be fitted by our N -body data. The explicit form ofthe fitting function is given in Appendix B.

3. SIMULATIONS

We use cosmological dark-matter N -body simulationsto calibrate the fitting formula. We use an N -bodydata set prepared by the Dark Emulator project(Nishimichi et al. 2019, hereafter N19). N19 prepared101 flat cosmological models (a fiducial ΛCDM with ad-ditional 100 wCDM models) in the range of z = 0–1.48.Their purpose is to construct an emulator for severalhalo observables such as halo-matter correlation func-tion, halo mass function and halo bias for ongoing weaklensing surveys. Their emulator will be publicly availablesoon.

3.1. Cosmological models

We use the N19 simulations for 41 flat cosmologicalmodels8. The fiducial ΛCDM model is consistent withthe Planck 2015 best-fit (Planck Collaboration 2016):the matter density Ωm = 1 − ΩΛ = 0.3156, baryon den-sity Ωb = 0.0492, Hubble constant h = 0.6727, spectralindex ns = 0.9645, and amplitude of matter density fluc-tuation at the scale of 8 h−1Mpc σ8 = 0.831.The other wCDM models have 6 cosmological param-

eters of Ωbh2,Ωcdmh

2,Ωw, As, ns and w. Here, the darkenergy equation of state, w, is assumed to be constant,and As is the amplitude of the primordial PS. These pa-rameters are distributed around the fiducial model in therange of ±5% for Ωbh

2 and ns, ±10% for Ωcdmh2, and

±20% for Ωw, lnAs and w. N19 sampled the cosmologi-cal parameters based on a Latin Hypercube Design (e.g.,Heitmann et al. 2009). They prepared 5 subsets contain-ing 20 models for each. We use their two subsets. Fig-ure 1 shows the distribution of w and σ8 vs. Ωm. Theparameter range is wide enough for the current and fu-ture weal-lensing surveys.These are dark-matter only simulations but the free-

streaming dumping by massive neutrinos is accounted forin the initial condition9. In all the models, the neutrinodensity is fixed to be Ωνh

2 = 6.4 × 10−4, correspondingto 0.06 eV for the total mass. This Ων is included in Ωm.

3.2. N -body simulations

Our simulation setting is summarized in Table 1. Weused four box sizes, L = 4, 2, 1, 0.2 h−1Gpc (side lengthsof the cubic boxes in comoving scale), to cover a widerange of length scales. Note that the simulation volume islarge enough to cover almost all measurable triangle con-figurations of BS in the real universe. The large-volumesimulations (> 10Gpc3) are required for a less noisy mea-surement of BS at small k. The small-box simulations(L = 0.2 h−1Gpc) will be used to check the asymptoticbehavior at high z. Here, the simulations with L = 1and 2 h−1Gpc at z = 0–1.48 are taken from N19, whilethe others are newly prepared in this work. The largest-and smallest-box simulations are performed to supple-ment the dynamic range covered by N19. The numberof particles is 40963 for L = 4 h−1Gpc and 20483 forthe rest. There are dozens of realizations for the fidu-cial ΛCDM model while there is a single one for eachwCDM model. The initial matter PS is prepared by

8 Unfortunately, the particle position data was lost for the rest(60) of the models due to hard-disk trouble.

9 They computed the linear transfer function at z = 0 with theneutrinos and then multiplied it by the linear growth factor squaredwithout the neutrinos to scale back to the initial redshift.

4 Takahashi et al.

TABLE 1N-body Simulation Parameters

Cosmological Box Size Number of Number of Maximum wavenumber Outputmodel (h−1 Gpc) particles realizations (hMpc−1) redshifts

Planck 2015 ΛCDM 4 40963 8 2.85 0, 0.55, 1.03, 1.48 & HighZ2 20483 15 1.42 LowZ1 20483 21 28.5 LowZ & HighZ0.2 20483 10 14.2 HighZ

40 wCDM 2 20483 1 1.42 LowZ1 20483 1 28.5 LowZ

Note. — In the column of output redshifts, LowZ means ten low redshifts: z = 0, 0.15, 0.31, 0.42, 0.55, 0.69,0.85, 1.03, 1.23, 1.48. HighZ is four high redshifts: z = 2, 3, 5, 10. The simulations with L = 1 and 2h−1 Gpc at LowZare taken from Nishimichi et al. (2019), while the others are newly prepared in this work.

the public Boltzmann code CAMB (Lewis et al. 2000).The initial particle distribution is prepared based onthe second order Lagrangian perturbation theory (2LPT;Crocce et al. 2006; Nishimichi et al. 2009)10 at redshiftszin = 31, 29, 59, 99 for L = 4, 2, 1, 0.2 h−1Gpc, respec-tively. We slightly change the initial redshifts for the 40wCDM models because the initial amplitude is slightlydifferent in each model. We then follow the non-lineargravitational evolution using Gadget2 (Springel et al.2001; Springel 2005). The particle snapshots are dumpedat 14 redshifts in z = 0–10 (see Table 1 for the exact out-put redshifts).In order to measure the density contrast, we assign the

N -body particles to 10243 regular grid cells in the box.Then, we perform the fast Fourier transform (FFT)11

to obtain its Fourier transform, δ(k). Here, we employthe CIC (cloud-in-cell) interpolation with the interlacingscheme (e.g., Jing 2005; Sefusatti et al. 2016). We alsoemploy the folding method (Jenkins et al. 1998) to ex-plore smaller scales by using 1/4 and 1/10 times smallerside lengths for L = 4 and 1 h−1Gpc, respectively. Theminimum and maximum wavenumbers with 10243 cellsare kLmin = 2π/L = 6.3× 10−3 hMpc−1 [L/(h−1Gpc)]−1

and kLmax = 512 kLmin = 3.2 hMpc−1 [L/(h−1 Gpc)]−1, re-spectively. After performing the folding scheme, bothkLmin and kLmax simply become 4 or 10 times larger. Theresultant kLmax is given in Table 1.

3.3. Power spectrum measurement for accuracy check

We measure the matter PS from the simulations andthen compare with the previous fitting formula to checkthe numerical accuracy. The PS estimator is given by,

P (k) =1

NPSmode

∑

|k′|∈k

∣

∣

∣δ(k′)

∣

∣

∣

2

, (5)

where the summation is done in a range of k −∆k/2 <|k′| < k+∆k/2 and NPS

mode is the number of mode withina fixed bin-width of ∆ log10 k = 0.1. Figure 2 plots a PSratio to the revised Halofit prediction (Smith et al. 2003;Takahashi et al. 2012). We plot average P (k) with 1σerror measured from the realizations. The results fromdifferent boxes nicely agree with each other. As the sim-ulation box becomes larger, the measured PS becomes

10 The 2LPT can reduce error of the BS estimate inducedby transients from initial conditions to less than 2% at z ≤ 1(McCullagh et al. 2016).

11 FFTW (Fast Fourier Transform in the West) is available athttp://www.fftw.org.

0.9

1

1.1

k (h/Mpc)

P /P

(k)

(k)

halofit

z=0 z=

0.55 z=1 z=

2 z=3

simulations L= Gpc/h0.21,2,4,

0.9

1

1.1

k (h/Mpc)

P /P

(k)

(k)

halofit

z=0 z=

0.55 z=1 z=

2 z=3


0.9

1

1.1

k (h/Mpc)

P /P

(k)

(k)

halofit

z=0 z=

0.55 z=1 z=

2 z=3


0.9

1

1.1

k (h/Mpc)

P /P

(k)

(k)

halofit

z=0 z=

0.55 z=1 z=

2 z=3


0.9

1

1.1

0.1 1 10

k (h/Mpc)

P /P

(k)

(k)

halofit

z=0 z=

0.55 z=1 z=

2 z=3


Fig. 2.— Power spectrum ratio of the simulation to the Halofitprediction for the Planck 2015 best-fit ΛCDM model. Each symbolcorresponds to each simulation box-size, L = 4, 2, 1, 0.2h−1 Gpc(blue, green, black, gray). Here, the shot noise contribution is lessthan 3%.

smaller than the Halofit prediction at large k due tothe lack of spacial resolution. Here, we do not subtractthe shot noise since its contribution is less than 3% atthe scales shown in the figure. The simulations agreewith the fitting formula within 5% for k < 1 hMpc−1 atz = 0–10 and for k < 10 hMpc−1 at z = 0–1.5.

3.4. Bispectrum measurement

http://www.fftw.org


The BS estimator is

B(k1, k2, k3) =1

Ntriangle

∑

|k′

1|∈k1

∑

|k′

2|∈k2

∑

|k′

3|∈k3

× δ(k′1)δ(k

′2)δ(k

′3)δ

Kk′

1+k′

2+k′

3

(6)

where the summation is done for all the modes in thebin, |k′

i| ∈ ki (i = 1, 2, 3), Ntriangle is the number oftriangles, and δK is the Kronecker delta. Throughoutthis paper, we adopt a constant bin-width in log scale,∆ log10 k = 0.1. Here, we employ the FFT-based quickestimator to calculate Eq. (6) (e.g., Scoccimarro 2015).

Using the identity, δKk′

1+k′

2+k′

3

= N−1cell

∑

xei(k

′

1+k

′

2+k

′

3)·x,

Eq. (6) can be reduced to,

B(k1, k2, k3) =1

Ntriangle

1

Ncell

∑

x

∑

|k′

1|∈k1

δ(k′1) e

ik′

1·x

×∑

|k′

2|∈k2

δ(k′2) e

ik′

2·x

∑

|k′

3|∈k3

δ(k′3) e

ik′

3·x

,

(7)

where x is the discrete coordinate on the grid, andNcell = 10243 is the total number of cells. The sum-mation over k′

i can be easily evaluated with FFT.We also measure the shot noise as,

Bsn(k1, k2, k3) =1

np

[

P (k1) + P (k2) + P (k3)]

−2

n2p

,

where np is the particle number density and P (k) is thePS estimator including the shot noise.For the fiducial model, we calculate the average and

standard deviation of BS from among the realizations(see Table 1 for the exact number of realizations). How-ever, since there is only a single realization in eachwCDM model, the result has a large scatter. Therefore,we will calibrate the fitting formula mainly for the Planck2015 model and further investigate the other models onlysupplementarily to check the dependence on the cosmo-logical parameters.

4. FITTING PROCEDURE AND RESULTS

4.1. Fitting to the N -body results

In this section, we present our fitting procedure to theN -body results. The BS fitting function in Eq. (4) con-tains 52 parameters. Denoting an array of these param-eters as p = (p1, p2, ...), we perform the standard chi-squared analysis to find the best fit:

χ2sim(p) =

∑

z,c,L

∑

k1,k2,k3

WcWzWk

×

[

Bbin(k1, k2, k3;p)−Bsim(k1, k2, k3)

∆Bsim(k1, k2, k3) + ǫ(k1, k2, k3)

]2

, (8)

where the summation is done for the 41 cosmologicalmodels labeled as c, all the redshifts z = 0–10, the simu-lations in different box sizes L, and triangles (k1, k2, k3)up to 30 hMpc−1. Here, Bbin is the binned prediction ofthe fitting formula (will be given in Eq. 9), Bsim is the

0.01

0.1

1

10

kmax (h/Mpc)

kmid

(h/M

pc)

kmin

(h/M

pc)

z=0.55

k mid=k m

ax

k mid=k m

ax/2

simulations L= 1Gpc/h2,4,1−loop SPT

k min=k m

ax

0.01

0.1

1

10

0.01 0.1 1 10

kmax (h/Mpc)

kmid

(h/M

pc)

kmin

(h/M

pc)

z=0.55

k mid=k m

ax

k mid=k m

ax/2

simulations L= 1Gpc/h2,4,1−loop SPT

k min=k m

ax

Fig. 3.— Triangle configurations included in the calibrationfrom the simulations with L = 4, 2, 1h−1 Gpc (blue, green andblack squares) and from the 1-loop standard perturbation theory(SPT; orange diamonds). Here, kmin, kmid and kmax are the mini-mum, middle and maximum side-length of a triangle. The dashedred lines correspond to particular triangles: kmid = kmax is thesqueezed, kmid = kmax/2 is the flattened, and kmin = kmax is theequilateral. The arrows in the upper panel indicate the maximumwavenumbers in the calibration from the simulations (blue, greenand black) and from the SPT (orange). The thick (thin) arrowsare with (without) the folding scheme. All the points satisfy theconditions a) – c) in section 4.1.

simulation result, and ∆Bsim is the standard deviationestimated from the N -body realizations. Since we haveonly one realization for each of the 40 wCDM models, weassume that the relative standard deviations, ∆lnBsim

(≡ ∆Bsim/Bsim), for these models are the same as thosefor the Planck 2015. We also include a “softening” termǫ = 0.02×Bsim to reduce an influence of some data points

6 Takahashi et al.

having very small ∆Bsim12. We introduce weight factors,

W , to give a greater importance to the data points (i)at lower redshift (Wz) because cosmic shear probes low-z (. 0.5) structures, (ii) at larger scale (Wk) becausethe unaccounted baryonic effects can play a role at smallscale (k & 1 hMpc−1), and (iii) at the fiducial cosmolog-ical model (Wc)

13.In this analysis, we include all the triangles (k1, k2, k3)

satisfying the following three conditions:

a) The relative standard deviation is smaller than10% (i.e., ∆lnBsim < 0.1).

b) The shot-noise contribution is less than 3%.

c) The larger-box simulation gives smaller Bsim athigh k due to the lack of spatial resolution (see alsoFig. 2 in the PS case). Therefore, if the deviationis larger than 3% with the small statistical error of∆lnBsim < 3%, we use the smaller-box result only.

The conditions a) and b) exclude data points at verysmall k and large k, respectively. The condition c) hasnegligible impact on the data selection. Figure 3 plotstriangles (k1, k2, k3) satisfying all the above conditionsfor L = 1, 2, 4 h−1Gpc at z = 0.55. In a range of0.1 . k/(hMpc−1) . 2, we can use the simulation re-sults from all the box-sizes, enabling us to give a reliablefit. This figure clearly shows that the simulations coveralmost all the triangles up to k = 3 hMpc−1. The lowerpanel shows a discontinuity at kmax ≃ 3 (0.8)hMpc−1 forL = 1 (4)h−1Gpc where the box size effectively changesfrom L to L/10 (L/4) due to the folding scheme (see alsosection 3.2). In the bottom panel, triangles in the lower-right part are missing, which means the calibration doesnot include very squeezed cases (kmax ≫ kmin). This isbecause the maximum value of kmax/kmin is 512 which isdetermined by the number of FFT grids (10243). The useof the folding method does not change this ratio (512).The number of independent triangular bins calibrated bythe simulations in each cosmological model and redshiftis about 950 at low z (= 0, 0.55, 1 and 1.48) and 690 athigh z (= 2, 3 and 5), respectively.One should consistently bin both the simulation re-

sult and the fitting formula when comparing them be-cause BS is sensitive to the binning especially for thesqueezed limit (Sefusatti et al. 2010; Namikawa et al.2019). Throughout this paper, we use the binned fittingformula:

Bbin(k1, k2, k3) =1

Ntriangle

∫

|k′

1|∈k1

d3k′1

∫

|k′

2|∈k2

d3k′2

∫

|k′

3|∈k3

d3k′3

×B(k′1, k′2, k

′3) δD(k

′1 + k′

2 + k′3),

(9)

12 Since the number of realizations are not large enough forestimating the variance accurately, some data points accidentallyhave very small ∆Bsim.

13 These weights are set as Wz = 8, 3, 1 and 0.3 for z ≤0.1, 0.1 < z ≤ 1, 1 < z ≤ 3 and z > 3; Wk = 3, 1 and 0.3for kmax/(hMpc−1) ≤ 3.2, 3.2 < kmax/(hMpc−1) ≤ 10, andkmax/(hMpc−1) > 10; Wc = 1 (8 × 10−4) for the Planck 2015(otherwise). These are chosen so as to achieve 10% accuracy ofthe fitting for kmax < 3hMpc−1 at z = 0–3 for the Planck 2015.

where B(k′1, k′2, k

′3) is the unbinned one and the number

of triangles is

Ntriangle =

∫

|k′

1|∈k1

d3k′1

∫

|k′

2|∈k2

d3k′2

∫

|k′

3|∈k3

d3k′3 δD(k′1 + k′

2 + k′3).

(10)

Here, ki is the weighted mean wavenumber defined aski =

∫

|k′

i|∈ki

d3k′i k′i × [

∫

|k′

i|∈ki

d3k′i ]−1. We show how the

binning affects BS in Appendix B. Note that, althoughthe unbinned triangle (k′1, k

′2, k

′3) satisfies the triangle

condition (i.e., |k′1 − k′2| < k′3 < k′1 + k′2), the bin center(k1, k2, k3) does not always satisfy this.

4.2. Fitting to perturbation theory

Since the simulation result is noisy at large scale,we also consider perturbation theory for the calibra-tion on the linear to quasi-linear scales. Here, we usethe 1-loop SPT (standard perturbation theory) whichincludes the tree level and next-to-leading order terms(e.g., Scoccimarro 1997; Scoccimarro et al. 1998). Thechi-square is defined analogously to the previous one inEq. (8):

χ2spt(p) = Wspt

∑

z,c

∑

k1,k2,k3

WzWc

×

[

B(k1, k2, k3;p)−Bspt(k1, k2, k3)

∆B(k1, k2, k3) + ǫ(k1, k2, k3)

]2

, (11)

where Bspt is the SPT prediction. Note that we do notneed to consider a binning for B or Bspt in this case. Weset ∆B = 0.5 |Bspt −Btree| and ǫ = 0.01×Bspt. We alsoset Wspt = 0.08 so as to give more weight to the calibra-tion from simulations14, Wc = 1 (3×10−5) for the Planck2015 (otherwise), and Wz is the same as in section 4.1.We include all the triangles (k1, k2, k3) satisfying thatBspt agrees with Btree within 5% and up to 0.3 hMpc−1,and thus the fitting is restricted to large scales. While wecan consider any unbinned triangular configurations forthe SPT calibration, we use the same bin central valuesof (k1, k2, k3) as those for the L = 4 h−1Gpc simulationwith bin-width ∆ log10 k = 0.1. Then, the calibrationcovers a range of 1.6 × 10−3 ≤ k/(h−1Mpc−1) < 0.3at z = 0–10. Figure 3 shows all the triangles used inthe calibration by SPT (orange diamonds). The aver-age number of triangles in each cosmological model andredshift is about 350.

4.3. Result

We introduce the total χ2,

χ2(p) = χ2sim(p) + χ2

spt(p), (12)

and then numerically search for the best-fitting param-eters p by minimizing χ2. The resulting best-fit modelis presented in Appendix B. We use the downhill sim-plex routine (amoeba) in Numerical Recipes (Press et al.2002) to find the minimum.Figure 4 plots the matter BS using the tree level for-

mula, our fitting formula, SC01, or GM12 comparedwith the simulation results for the Planck 2015 model

14 The resulting χ2sim is about 40 times larger than χ2

spt.


105

106

B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)

equilateral flattened squeezedm

atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2

BiHalofitGM12SC01 tree

simulationsL= 1, Gpc/h2,4, 0.2

B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



107

108

B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



104

105

106

B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



106

107

B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



104

105

B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



106

107

B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



103

104

0.1 1 10

B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



0.1 1 10

B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



105

106

0.1 1

B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



1 10

B(k,k,k)

×k3

B(k,k/2,k/2)

×0.2k3

B(k,k,0.045)

×kB(k,k,0.45)

×10k2

k (h/Mpc)


atte

r bi

spec

trum

z=0 z=

0.55 z=1 z=

2



Fig. 4.— Matter bispectrum comparison of N-body simulations and fitting formulas for the Planck 2015 best-fit ΛCDM model. The curvesdenote theoretical models: our fit (BiHalofit; solid red), Gil-Marın et al. (2012) (GM12; long-dashed orange), Scoccimarro & Couchman(2001) (SC01; short-dashed pink) and the tree level (dotted purple). The symbols denote the simulation results with various box-sizes:L = 4, 2, 1h−1 Gpc (blue triangle, green triangle, black circle) and 200h−1 Mpc (gray diamond). Both theories and simulations areconsistently binned with ∆ log10 k = 0.1. In the vertical axis, the bispectrum is multiplied by ∝ kn (n = 1, 2 or 3) as denoted above thetop panels for a clear presentation. Throughout this paper, k and B(k1, k2, k3) have the units of hMpc−1 and (h−1 Mpc)6, respectively.

8 Takahashi et al.

0.8

0.9

1

1.1

1.2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)

equilateral flattened squeezed

ratio

z=0 z=

0.55 z=1 z=

2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

0.8

0.9

1

1.1

1.2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

0.8

0.9

1

1.1

1.2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

0.8

0.9

1

1.1

1.2

0.1 1 10

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

0.1 1 10

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

0.1 1

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

1 10

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045) B(k,k,0.45)

k (h/Mpc)


ratio

z=0 z=

0.55 z=1 z=

2

Fig. 5.— Same as Fig. 4, but ratios to the red curves (BiHalofit).


1

2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045)

k (h/Mpc)


B/B

tree

z=0 z=

0.55 z=1 z=

2

0.2 0.5 0.5

BiHalofit1−loop SPT

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045)

k (h/Mpc)


B/B

tree

z=0 z=

0.55 z=1 z=

2

0.2 0.5 0.5


B(k,k,k) B(k,k/2,k/2) B(k,k,0.045)

k (h/Mpc)


B/B

tree

z=0 z=

0.55 z=1 z=

2

0.2 0.5 0.5


1

1.5

2

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045)

k (h/Mpc)


B/B

tree

z=0 z=

0.55 z=1 z=

2

0.2 0.5 0.5


B(k,k,k) B(k,k/2,k/2) B(k,k,0.045)

k (h/Mpc)


B/B

tree

z=0 z=

0.55 z=1 z=

2

0.2 0.5 0.5


B(k,k,k) B(k,k/2,k/2) B(k,k,0.045)

k (h/Mpc)


B/B

tree

z=0 z=

0.55 z=1 z=

2

0.2 0.5 0.5


1

1.4

1.8

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045)

k (h/Mpc)


B/B

tree

z=0 z=

0.55 z=1 z=

2

0.2 0.5 0.5


B(k,k,k) B(k,k/2,k/2) B(k,k,0.045)

k (h/Mpc)


B/B

tree

z=0 z=

0.55 z=1 z=

2

0.2 0.5 0.5


B(k,k,k) B(k,k/2,k/2) B(k,k,0.045)

k (h/Mpc)


B/B

tree

z=0 z=

0.55 z=1 z=

2

0.2 0.5 0.5


1

1.2

1.4

0.1

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045)

k (h/Mpc)


B/B

tree

z=0 z=

0.55 z=1 z=

2

0.2 0.5 0.5


0.1

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045)

k (h/Mpc)


B/B

tree

z=0 z=

0.55 z=1 z=

2

0.2 0.5 0.5


0.1

B(k,k,k) B(k,k/2,k/2) B(k,k,0.045)

k (h/Mpc)


B/B

tree

z=0 z=

0.55 z=1 z=

2

0.2 0.5 0.5


Fig. 6.— Similar to Fig. 5, but ratios to the tree level, focusing on quasi-linear scales. The dot-dashed orange curves are the binned1-loop SPT prediction.

10 Takahashi et al.

0.8

0.9

1

1.1

1.2

kmax (h/Mpc)

ratio

z=0 z=

0.55 z=1 z=

2 z=3

simulations L= 1, Gpc/h0.22,4,

0.8

0.9

1

1.1

1.2

kmax (h/Mpc)

ratio

z=0 z=

0.55 z=1 z=

2 z=3


0.8

0.9

1

1.1

1.2

kmax (h/Mpc)

ratio

z=0 z=

0.55 z=1 z=

2 z=3


0.8

0.9

1

1.1

1.2

kmax (h/Mpc)

ratio

z=0 z=

0.55 z=1 z=

2 z=3


0.8

0.9

1

1.1

1.2

0.1 1 10

kmax (h/Mpc)

ratio

z=0 z=

0.55 z=1 z=

2 z=3


Fig. 7.— Bispectrum ratios of the simulations to BiHalofit for all triangles for the Planck 2015 model. The horizontal axis is a maximumwave number of (k1, k2, k3). Dozens of points are distributed along vertical axis on each kmax. Here, horizontal positions for different Lare slightly offset for a clear presentation.


0.6

0.8

1

1.2

1.4

kmax (h/Mpc)

ratio

z=0 z=

0.55 z=1 z=

1.48

S8<0.6, S8=0.6−1.0, S8>1.0

0.6

0.8

1

1.2

1.4

kmax (h/Mpc)

ratio

z=0 z=

0.55 z=1 z=

1.48

S8<0.6, S8=0.6−1.0, S8>1.0

0.6

0.8

1

1.2

1.4

kmax (h/Mpc)

ratio

z=0 z=

0.55 z=1 z=

1.48

S8<0.6, S8=0.6−1.0, S8>1.0

0.6

0.8

1

1.2

1.4

0.1 1

kmax (h/Mpc)

ratio

z=0 z=

0.55 z=1 z=

1.48

S8<0.6, S8=0.6−1.0, S8>1.0

Fig. 8.— Same as Fig. 7, but for the 40 wCDM models at z = 0–1.48. Each color corresponds to each cosmological model shown inFig. 1. The cyan, gray and magenta points have different S8 ≡ σ8(Ωm/0.3)0.5 values. The plus (cross) symbols are simulations withL = 1 (2) h−1 Gpc. Note that every data point has intrinsic scatter of 10% level because there is a only single realization in each wCDMmodel.

at z = 0–2. The four panels, from left to right, corre-spond to particular triangle configurations: the equilat-eral (i.e., k1 = k2 = k3), flattened (k1 = 2k2 = 2k3) andtwo squeezed cases (k1 = k2 ≫ k3 with k3 = 0.045 and0.45 hMpc−1). In this and following figures, the simu-lation data points satisfy all the conditions a) – c) insection 4.1. Here, for SC01 and GM12, we used the mea-sured PS in their fitting formulae to remove the inac-curacy of the PS appearing in these models. Figure 5plots ratios of the different models and simulations toour fitting formula. As clearly seen, our fitting formulais in good agreement with the simulations over the scales,redshifts, and triangle shapes. On the other hand, the

previous formulae over-predict the squeezed BS, which isconsistent with the previous finding by Namikawa et al.(2019). This figure also shows agreement among the sim-ulations performed in different box sizes.Figure 6 shows BS ratios to the tree-level at quasi non-

linear scales. On larger scales, both our simulations andfitting formula are consistent with the tree level predic-tion. The 1-loop SPT slightly over-predicts the BS atquasi non-linear scales at low z (< 1) (consistent withFig. 19 of Lazanu et al. 2016). The SPT is more ac-curate at higher redshifts. For the flattened case, theSPT slightly suppresses the BS at k ∼ 0.1 hMpc−1 andour model captures this trend. Some data points at

12 Takahashi et al.

k < 0.1 hMpc−1 are missing because of a large relativeerror (> 10%).Figure 7 shows the BS ratios to our formula for all

the triangles satisfying the conditions a) – c) in section4.1. The x-axis is the maximum wavenumber kmax andthere are ∼ 1800 data points in each redshift. Our modelagrees with the simulations within 10 (15)% accuracy upto k = 3 (10)hMpc−1 at z = 0–3. At z = 5 (10), theagreement is 20% level up to k = 3 (1)hMpc−1. Ther.m.s. deviation is 2.7 (3.2)% up to k = 3 (10)hMpc−1 atz = 0–3. We have confirmed that the accuracy does notdepend on the bin width by explicitly testing a narrowerbin width, ∆ log10 k = 0.05. In this case, the r.m.s. devi-ation is 2.9 (3.4)% up to k = 3 (10)hMpc−1 at z = 0–3,which is quantitatively consistent with those above.Figure 8 plots the BS ratios to our model for the 40

wCDM models. In this case, we prepare a single realiza-tion for each cosmological model and box size (L = 1 and2 h−1Gpc), and therefore the BS measurement has rela-tively large scatter of typically 10% level. Here, all thedata points satisfy the conditions a) – b) in section 4.1.There are huge number of data points (∼ 5 × 104) ineach redshift. The r.m.s. deviation is 8.0 (11.2)% up tok = 3 (10)hMpc−1 at z = 0–1.5.To further investigate the cosmology dependence of the

accuracy, we divide the models into three groups basedon the value of S8 = σ8(Ωm/0.3)0.5. The data pointsshown in Fig. 8 are color coded in the same manner asin Fig. 1. Our formula shows a good agreement withinabout 20% for the models with S8 = 0.6–1.0. Howeverthe agreement is worse for smaller or larger S8 models,because the amplitude of fluctuations (σ8) and the lineargrowth factor are largely different from the Planck 2015best-fit model. As all the cosmological models convergeto the Einstein–de Sitter at high z, the fit is better athigher redshifts.

5. BARYONIC EFFECTS

Our N -body simulations do not include baryonic pro-cesses such as gas cooling, star formation, supernovaeand active galactic nuclei (AGN) feedbacks. As wellrecognized, baryons significantly affect the non-linearPS at k & 1 hMpc−1 (e.g., van Daalen et al. 2011;Semboloni et al. 2011a; Osato et al. 2015; Hellwing et al.2016; Chisari et al. 2018, 2019). In this section, wetake into account the baryonic effects on the BS fit-ting formula using state-of-art hydrodynamic simula-tions, the IllustrisTNG data set15 (Nelson et al. 2019;Springel et al. 2018). The simulations follow galaxy for-mation and evolution by incorporating astrophysical pro-cesses with a subgrid model. In the IllustrisTNG project,three sets of simulations in different box sizes are car-ried out with three mass resolutions in each box-size.Here, we use the highest-resolution one in the largestbox (referred to as TNG300-1), in which the box-sizeis L = 205 h−1Mpc (≃ 300Mpc) with 25003 dark mat-ter and baryon particles, respectively. Their cosmolog-ical model is based on the Planck 2015 best-fit ΛCDM(Planck Collaboration 2016). They released particle po-sitions and masses of dark matter and baryons (in the

15 http://www.tng-project.org

form of gas, star and black hole) at z = 0–20. The Illus-trisTNG team also performed dark-matter only (dmo)runs. A comparison between the simulations with andwithout baryons enables us to single out the impact ofbaryons on the matter clustering.We assign all the particle masses to 10243 grid cells in

order to calculate the density contrast and then measurethe BS following the same procedure as in section 3.4.We define a BS ratio of the simulations with baryons(Bb) to the dmo run (Bdmo) as,

Rb(k1, k2, k3) =Bb(k1, k2, k3)

Bdmo(k1, k2, k3). (13)

We measure this ratio at nine redshifts of z = 0–5 (z =0, 0.2, 0.4, 0.7, 1, 1.5, 2, 3, and 5).Figure 9 plots the ratios for three triangle configu-

rations at z = 0–2. The simulations with and with-out baryons have the same seed in their initial condi-tions, which reduces the sample-variance scatter in theratio at large scales. The baryons clearly suppress thepower due to the AGN feedback at k ∼ 10 hMpc−1, butstrongly enhance the amplitude due to gas cooling athigh k (> 10 hMpc−1). This trend is consistent with thePS (see also Fig. 10). However, the baryons slightly en-hance the amplitude by about 10% at intermediate scale(k ≃ 1–10 hMpc−1) only for low redshifts (z < 1). Toour knowledge, this small enhancement is not commonin PS.Figure 10 plots the PS ratio with and without baryons

measured from the TNG300-1. The circles (crosses) areratios of the total matter (dark-matter) PS to the dmorun. The small enhancement at intermediate scales ap-pears in the crosses but can not be seen in the cir-cles. This feature was also mentioned in section 3 ofSpringel et al. (2018). Therefore, the dark-matter com-ponent causes this enhancement. We can see that theenhancement of the dark-matter PS appears at almostthe same wavenumbers as in the total-matter BS.While preparing for this paper, Foreman et al. (2019)

posted a paper on arXiv about the baryonic effects onBS measured from hydrodynamic simulations (includingthe TNG300-1). They also found the same trend andgave a clear explanation of what causes this. After theAGN feedback becomes less effective at late time (z <1), the expelled gas re-accretes to a halo. Then the gascontraction affects the dark matter distribution in thehalo. As the BS is more sensitive to the dark matter (notbaryons) compared to PS (see their section 3.1.1), thisenhancement appears only in BS. The baryonic effects onboth PS and BS can be used to discriminate the baryonicmodels (Semboloni et al. 2013; Foreman et al. 2019).To incorporate the baryonic effect to our BS model,

we also construct a fitting function of the ratio, Rb inEq. (13), shown as the solid red curves in Fig. 9. Its ex-plicit functional form is given in Appendix C. This can fitthe measurement within 7.3 (5.3)% for k < 10 hMpc−1

at low (high) redshift, z = 0–1 (1.5–5). The r.m.s. de-viation is 1.9% for k < 30 hMpc−1 at z = 0–2. Thenumber of triangles in this fit is about 760 (6800) in eachredshift (all the redshifts of z = 0–5). The user can eas-ily include the baryonic effects by multiplying this ratioby the BS fitting formula. This approach is the same as

http://www.tng-project.org


0.6

0.8

1

1.2

1.4

1.6

1 10

(k,k,k) (k,k/2,k/2) (k,k,1.08)

k (h/Mpc)

equilateral flattened squeezedR

b(k

1,k2,k3) z=0

0.412

0.8

1

1.2

1 10

(k,k,k) (k,k/2,k/2) (k,k,1.08)

k (h/Mpc)


b(k

1,k2,k3) z=0

0.412

0.8

1

1.2

1.4

1 10

(k,k,k) (k,k/2,k/2) (k,k,1.08)

k (h/Mpc)


b(k

1,k2,k3) z=0

0.412

Fig. 9.— Ratio of bispectrum with to without baryons, Rb = Bb/Bdmo defined in Eq. (13), measured from the TNG300-1. The filledcircles are the total-matter (dark matter and baryons) bispectrum divided by that from the dark-matter only run. The red curves are ourfit given in Appendix C.

0.8

1

1.2

1 10 100

k (h/Mpc)

P

/P

(k)

(k)

b

dmo

z=00.412

Fig. 10.— Similar to Fig. 9, but ratio of P (k) measured from theTNG300-1. The filled circles (crosses) are the total-matter (darkmatter) P (k) divided by that from the dark-matter only run.

Harnois-Deraps et al. (2015) done for baryonic effects onPS.

6. COMPARISON WITH WEAK LENSING SIMULATIONS

With the fitting formula for the matter BS calibratedover a wide wavenumber and redshift range, we can makea prediction for the lensing observable by integratingalong the line of sight. In this section, we compare ourtheoretical prediction with weak lensing BS measuredfrom ray-tracing simulations. Here we consider the con-vergence BS for CMB lensing (section 6.1) and cosmicshear (section 6.2).The convergence field is a dimensionless integrated

matter density along the line-of-sight direction towardsthe source. The convergence at an angular posi-

tion θ for a source distance rs is given by (e.g.,Bartelmann & Schneider 2001),

κ(θ) =

∫ rs

0

drW (r, rs) δ(rθ, r; z), (14)

with the weight function

W (r, rs) =3H2

0Ωm

2c2r (rs − r)

a(r) rs, (15)

where r (rs) is the comoving distance (to the source) anda(r) is the scale factor. The convergence BS is

Bκ(ℓ1, ℓ2, ℓ3) =

∫ rs

0

drW 3(r, rs)

r4B

(

ℓ1r,ℓ2r,ℓ3r; z

)

, (16)

where ℓi (= kir) is the multipole moment andB(k1, k2, k3; z) is the matter BS at z. This formula isderived under the flat-sky and the Born approximations.For a higher source redshift, the Born approximation isless accurate and we need the so-called post-Born cor-rections (Pratten & Lewis 2016). We include these cor-rections only for CMB lensing (safely neglected for cos-mic shear because its contribution is O(1%), see Fig.7 ofPratten & Lewis 2016).The convergence BS is sensitive to lower-z structures

compared to the convergence PS for a given source red-shift (e.g., see Fig. 4 of Takada & Jain 2002), becausethe matter BS (PS) evolves in proportional to the forthpower (square) of the linear growth factor in the linearregime. Therefore, they can probe different-z structuresin a complimentary manner.

6.1. CMB lensing

Namikawa et al. (2019) recently measured the con-vergence BS from full-sky light-cone simulations(Takahashi et al. 2017). Here, we compare their mea-surement with theoretical predictions. Takahashi et al.(2017) ran cosmological N -body simulations to repro-duce inhomogeneous mass distribution in the universe

14 Takahashi et al.

0

0.5

1

1.5

Bκ(ℓ,ℓ,ℓ)

×10 ℓ8 2.5

Bκ(ℓ,ℓ/2,ℓ/2)

×10 ℓ9 2

Bκ(ℓ,ℓ,50)

×10 ℓ10 1.5

Bκ(ℓ,1000,1000)

×10 ℓ11 1.5

ℓ

equilateral flattened squeezed isoscelesco

nver

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∆B/σ

0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000

simulation

BiHalofitGM12SC01

CMB lensing

Bκ(ℓ,ℓ,ℓ)

×10 ℓ8 2.5

Bκ(ℓ,ℓ/2,ℓ/2)

×10 ℓ9 2

Bκ(ℓ,ℓ,50)

×10 ℓ10 1.5

Bκ(ℓ,1000,1000)

×10 ℓ11 1.5

ℓ


nver

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spec

trum

∆B/B

∆B/σ

0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000

simulation

BiHalofitGM12SC01

CMB lensing

Bκ(ℓ,ℓ,ℓ)

×10 ℓ8 2.5

Bκ(ℓ,ℓ/2,ℓ/2)

×10 ℓ9 2

Bκ(ℓ,ℓ,50)

×10 ℓ10 1.5

Bκ(ℓ,1000,1000)

×10 ℓ11 1.5

ℓ


nver

genc

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spec

trum

∆B/B

∆B/σ

0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000

simulation

BiHalofitGM12SC01

CMB lensing

Bκ(ℓ,ℓ,ℓ)

×10 ℓ8 2.5

Bκ(ℓ,ℓ/2,ℓ/2)

×10 ℓ9 2

Bκ(ℓ,ℓ,50)

×10 ℓ10 1.5

Bκ(ℓ,1000,1000)

×10 ℓ11 1.5

ℓ


nver

genc

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spec

trum

∆B/B

∆B/σ

0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000

simulation

BiHalofitGM12SC01

CMB lensing

−0.2

0

Bκ(ℓ,ℓ,ℓ)

×10 ℓ8 2.5

Bκ(ℓ,ℓ/2,ℓ/2)

×10 ℓ9 2

Bκ(ℓ,ℓ,50)

×10 ℓ10 1.5

Bκ(ℓ,1000,1000)

×10 ℓ11 1.5

ℓ


nver

genc

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spec

trum

∆B/B

∆B/σ

0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000

simulation

BiHalofitGM12SC01

CMB lensing

−0.4

0

Bκ(ℓ,ℓ,ℓ)

×10 ℓ8 2.5

Bκ(ℓ,ℓ/2,ℓ/2)

×10 ℓ9 2

Bκ(ℓ,ℓ,50)

×10 ℓ10 1.5

Bκ(ℓ,1000,1000)

×10 ℓ11 1.5

ℓ


nver

genc

e bi

spec

trum

∆B/B

∆B/σ

0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000

simulation

BiHalofitGM12SC01

CMB lensing

Bκ(ℓ,ℓ,ℓ)

×10 ℓ8 2.5

Bκ(ℓ,ℓ/2,ℓ/2)

×10 ℓ9 2

Bκ(ℓ,ℓ,50)

×10 ℓ10 1.5

Bκ(ℓ,1000,1000)

×10 ℓ11 1.5

ℓ


nver

genc

e bi

spec

trum

∆B/B

∆B/σ

0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000

simulation

BiHalofitGM12SC01

CMB lensing

Bκ(ℓ,ℓ,ℓ)

×10 ℓ8 2.5

Bκ(ℓ,ℓ/2,ℓ/2)

×10 ℓ9 2

Bκ(ℓ,ℓ,50)

×10 ℓ10 1.5

Bκ(ℓ,1000,1000)

×10 ℓ11 1.5

ℓ


nver

genc

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spec

trum

∆B/B

∆B/σ

0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000

simulation

BiHalofitGM12SC01

CMB lensing

Bκ(ℓ,ℓ,ℓ)

×10 ℓ8 2.5

Bκ(ℓ,ℓ/2,ℓ/2)

×10 ℓ9 2

Bκ(ℓ,ℓ,50)

×10 ℓ10 1.5

Bκ(ℓ,1000,1000)

×10 ℓ11 1.5

ℓ


nver

genc

e bi

spec

trum

∆B/B

∆B/σ

0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000

simulation

BiHalofitGM12SC01

CMB lensing

Bκ(ℓ,ℓ,ℓ)

×10 ℓ8 2.5

Bκ(ℓ,ℓ/2,ℓ/2)

×10 ℓ9 2

Bκ(ℓ,ℓ,50)

×10 ℓ10 1.5

Bκ(ℓ,1000,1000)

×10 ℓ11 1.5

ℓ


nver

genc

e bi

spec

trum

∆B/B

∆B/σ

0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000

simulation

BiHalofitGM12SC01

CMB lensing

Bκ(ℓ,ℓ,ℓ)

×10 ℓ8 2.5

Bκ(ℓ,ℓ/2,ℓ/2)

×10 ℓ9 2

Bκ(ℓ,ℓ,50)

×10 ℓ10 1.5

Bκ(ℓ,1000,1000)

×10 ℓ11 1.5

ℓ


nver

genc

e bi

spec

trum

∆B/B

∆B/σ

0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000

simulation

BiHalofitGM12SC01

CMB lensing

Bκ(ℓ,ℓ,ℓ)

×10 ℓ8 2.5

Bκ(ℓ,ℓ/2,ℓ/2)

×10 ℓ9 2

Bκ(ℓ,ℓ,50)

×10 ℓ10 1.5

Bκ(ℓ,1000,1000)

×10 ℓ11 1.5

ℓ


nver

genc

e bi

spec

trum

∆B/B

∆B/σ

0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000

simulation

BiHalofitGM12SC01

CMB lensing

Fig. 11.— Convergence bispectrum measured from simulation maps for CMB lensing. The black symbols are the averages with thestandard deviations measured from 108 full-sky maps (Takahashi et al. 2017; Namikawa et al. 2019). Here, the error bars scale as

[(survey area)/(4π)]−1/2. The solid red, dashed orange and dotted pink curves are theoretical predictions based on BiHalofit, GM12,and SC01, respectively. The middle panels plot relative deviations from the red curves: ∆B/B ≡ Bκ/BBiHalofit

κ − 1. In the bottom panel,these deviations are further divided by the relative standard deviation (σ/B).

from the present to the last scattering surface. The cos-mological model is consistent with the WMAP 9yr re-sult (Hinshaw et al. 2013). The authors performed aray-tracing simulation to calculate light-ray paths de-flected by the intervening matter. The light-rays areemitted from the position of an observer (at z = 0)and trace back the trajectories up to the last scatter-ing surface (for detailed discussion about the ray-tracingscheme, see Shirasaki et al. 2015). They do not usethe Born approximation and therefore it includes thepost-Born effects. They provide 108 full-sky conver-gence maps16 based on the HEALPix pixelization withNside = 8192 (4096), corresponding to a pixel size of0.48 (0.96) arcmin (Gorski et al. 2005). They confirmedthat the convergence PS agrees with theoretical predic-tion by CAMB with the Halofit PS option within 5% atℓ ≤ 2000 for the high-resolution maps (Nside = 8192).Figure 11 plots the BS measurement from the 108

maps with Nside = 8192 (Namikawa et al. 2019). Thecurves show theoretical predictions including the post-Born correction adopting the WMAP 9yr cosmologicalmodel consistently with the simulations. For a fair com-parison, both theoretical predictions and simulation re-sults are binned in the same manner with a bin-width

16 These maps are available athttp://cosmo.phys.hirosaki-u.ac.jp/takahasi/allsky_raytracing.

∆ℓ = 100. The error bars are for an ideal full-sky mea-surement (i.e., the cosmic-variance limit) and scale as[(survey area)/(4π)]−1/2∆ℓ−3/2 assuming the Gaussianvariance. Overall, our fitting formula provides better pre-dictions for the BS of the CMB lensing than the otherexisting formulae. In the equilateral case, the analyticand simulated BS agree within ∼ 10% accuracy at mostof the angular scales. The difference is within 0.2σ, asshown in the bottom panel. For the flattened case, theratio (in the middle panel) is far from unity because theBS is around zero at ℓ & 1000. The discrepancy is about0.2σ of the cosmic variance. In the squeezed and isoscelesconfigurations, our fitting formula significantly reducesthe discrepancy between the simulation and analyticalprediction.Although the accuracy is surely improved, there still

remains noticeable discrepancy between our fitting for-mula and the simulations. One of the reasons would bethe finite thickness of the lens planes employed in theray-tracing simulations which may affect the simulationsat ℓ < 200 (for the same effect on convergence PS, seeFig. 10 of Takahashi et al. 2017). Another reason wouldbe that the flat-sky formula in Eq. (16) becomes inaccu-rate for large angular scales (the accuracy of the flat-skyapproximation in the cosmic-shear PS is discussed in de-tail by Kilbinger et al. 2017; Kitching et al. 2017). For

http://cosmo.phys.hirosaki-u.ac.jp/takahasi/allsky_raytracing


example, in the squeezed limit, the minimum multipoleis fixed to be ℓ3 = 50, while the discrepancy is mitigatedif we choose a larger ℓ3 (Namikawa et al. 2019). Sincereducing the finite thickness of lens planes requires morenumerically expensive simulations, we will leave the de-tailed study for our future work.

6.2. Cosmic shear

Let us now turn to the cosmic-shear signal from mea-surements of galaxy shapes, which probes lower red-shifts compared to the CMB lensing. Sato et al. (2009)ran cosmological N -body simulations and subsequentlyperformed a ray-tracing simulation under the flat-skyapproximation. Their field of view is somewhat small(5×5 deg2) but the number of weak lensing maps (1000)is sufficient for an accurate measurement of BS. Theircosmological model is consistent with the WMAP 3yrresult (Spergel et al. 2007). Figure 12 plots the conver-gence BS at a source redshift zs = 1 measured from the1000 maps by Kayo et al. (2013). Both theory and sim-ulation are binned in the same manner with ∆ log10 ℓ =0.13. The simulation result is valid within 5% up toℓ ≃ 4000 confirmed by a comparison with low- and high-resolution maps (Sato et al. 2009; Valageas et al. 2012).Our fitting formula agrees with the simulation well within10% level up to ℓ = 4000. The deviation at small scale(ℓ & 4000) is due to the lack of resolution in the simula-tion.

7. DISCUSSION

7.1. Systematics in CMB lensing

In CMB lensing measurements, the lensing map is re-constructed through mode-mixing of CMB anisotropiesinduced by lensing (Hu & Okamoto 2002). There-fore, any other sources of mode-mixing could biaslensing measurements and thus the BS of CMB lens-ing. For example, masking, inhomogeneous noise, beamand point sources are potential sources of the bias(Hanson et al. 2009; Namikawa et al. 2013). In additionto the instrumental uncertainties, extra-galactic fore-grounds such as the thermal Sunyaev-Zel’dovich effectand cosmic infrared background (Osborne et al. 2014;van Engelen et al. 2014; Madhavacheril & Hill 2018) andits lensing (Mishra & Schaan 2019) could lead to the biasin CMB lensing measurements. The calibration uncer-tainties of the CMB map is also an important systematicsbecause the measured BS depends on the sixth power ofthe map calibration uncertainties if the quadratic estima-tor is used for lensing reconstruction. This dependenceis stronger than that of the lensing PS which dependson the forth power of these uncertainties. However, thebias from the above instrumental uncertainties and as-trophysical sources would be constrained by combiningthe BS and PS of CMB lensing since these spectra havea different dependence on these uncertainties. A jointanalysis of PS and BS would be thus crucial for a robustcosmological analysis in future CMB experiments.

7.2. Intrinsic alignment

The intrinsic alignment (IA) of galaxies is one of ma-jor systematics in cosmic shear (for a review, see e.g.,Troxel & Ishak 2015; Joachimi et al. 2015). A massive

structure near the source galaxy induces a shape distor-tion due to the tidal force, which becomes a contamina-tion to lensing signal. The IA contamination is about10% level for the cosmic-shear BS (Semboloni et al.2008). Several authors proposed a method to mitigate orto remove this contamination from the signal (Shi et al.2010; Troxel & Ishak 2012). Combining lensing PS andBS can provide a strong constraint on IA model as wellas cosmological parameters.

7.3. Bispectrum covariance

We have so far discussed the modeling of BS, butits covariance is also an important ingredient for acosmological likelihood analysis. For Gaussian fluc-tuations, the BS covariance has a simple form givenby the PS and the shot noise (Sefusatti et al. 2006).However, in the non-linear regime, one should considerthe non-Gaussian and super-sample contributions (e.g.,Takada & Hu 2013) and therefore its evaluation becomesmuch more difficult. In that case, several authors es-timated the covariance via perturbation theory (e.g.,Sugiyama et al. 2019), the halo model (e.g., Kayo et al.2013; Rizzato et al. 2018) and an ensemble of simulationmocks (e.g., Sato & Nishimichi 2013; Chan & Blot 2017;Chan et al. 2018; Colavincenzo et al. 2019). In the lastapproach, the number of mocks should be larger thana number of k-bins to estimate an unbiased inverse co-variance (e.g., Hartlap et al. 2007), and therefore one re-quires a huge number of mocks (> 102–3). Anyway, weleave this topic for future work.

7.4. Emulator

Several groups recently have been developing a non-linear PS emulator which interpolates simulation re-sults in a wide range of wavenumber, redshift, and cos-mological models (Lawrence et al. 2017; Garrison et al.2018; Nishimichi et al. 2019; Knabenhans et al. 2019;DeRose et al. 2019). We expect that it is probably muchmore difficult to construct a similar emulator for BS. Onereason is that we measure a binned BS while we need anunbinned one (note again that BS is sensitive to binning).Therefore, we cannot simply interpolate the measuredquantities. The other reason is that a BS measurementis quite noisy due to the larger sample variance comparedto PS, which means many realizations are needed in eachcosmological model. This is computationally expensive.

8. CONCLUSION

We have constructed a fitting formula of the matter BScalibrated by high-resolution N -body simulations for 41wCDM models around the Planck 2015 best-fit ΛCDMmodel. We also include a calibration from perturbationtheory at large scale (k < 0.3 hMpc−1). Our formulacan be used for a wide range of wavenumbers (up tok = 30 hMpc−1) and redshifts (z = 0–10). The simu-lation boxes are large enough (L = 1, 2, and 4 h−1Gpcon a side length) to cover almost all triangles (k1, k2, k3)measured by the forthcoming weak-lensing surveys andCMB lensing experiments. It achieves an accuracy of10 (15)% up to k = 3 (10)hMpc−1 at z = 0–3 for thePlanck 2015 model. The accuracy for the 40 wCDMmodels is about 20% level for k < 3 hMpc−1 at z = 0–1.5

16 Takahashi et al.

10−9

10−8

Bκ(ℓ,ℓ,ℓ)

×ℓ3

Bκ(ℓ,ℓ/2,ℓ/2)

×0.2ℓ3

Bκ(ℓ,ℓ,84)

×70ℓ2

Bκ(ℓ,ℓ,510)

×300ℓ2

ℓ

equilateral flattened squeezedco

nver

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spec

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∆B/B

simulationBiHalofit

cosmic shearzs=1

Bκ(ℓ,ℓ,ℓ)

×ℓ3

Bκ(ℓ,ℓ/2,ℓ/2)

×0.2ℓ3

Bκ(ℓ,ℓ,84)

×70ℓ2

Bκ(ℓ,ℓ,510)

×300ℓ2

ℓ


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simulationBiHalofit

cosmic shearzs=1

Bκ(ℓ,ℓ,ℓ)

×ℓ3

Bκ(ℓ,ℓ/2,ℓ/2)

×0.2ℓ3

Bκ(ℓ,ℓ,84)

×70ℓ2

Bκ(ℓ,ℓ,510)

×300ℓ2

ℓ


nver

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∆B/B

simulationBiHalofit

cosmic shearzs=1

Bκ(ℓ,ℓ,ℓ)

×ℓ3

Bκ(ℓ,ℓ/2,ℓ/2)

×0.2ℓ3

Bκ(ℓ,ℓ,84)

×70ℓ2

Bκ(ℓ,ℓ,510)

×300ℓ2

ℓ


nver

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simulationBiHalofit

cosmic shearzs=1

−0.2−0.1

0 0.1

100 1000

Bκ(ℓ,ℓ,ℓ)

×ℓ3

Bκ(ℓ,ℓ/2,ℓ/2)

×0.2ℓ3

Bκ(ℓ,ℓ,84)

×70ℓ2

Bκ(ℓ,ℓ,510)

×300ℓ2

ℓ


nver

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simulationBiHalofit

cosmic shearzs=1

100 1000

Bκ(ℓ,ℓ,ℓ)

×ℓ3

Bκ(ℓ,ℓ/2,ℓ/2)

×0.2ℓ3

Bκ(ℓ,ℓ,84)

×70ℓ2

Bκ(ℓ,ℓ,510)

×300ℓ2

ℓ


nver

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simulationBiHalofit

cosmic shearzs=1

100 1000

Bκ(ℓ,ℓ,ℓ)

×ℓ3

Bκ(ℓ,ℓ/2,ℓ/2)

×0.2ℓ3

Bκ(ℓ,ℓ,84)

×70ℓ2

Bκ(ℓ,ℓ,510)

×300ℓ2

ℓ


nver

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simulationBiHalofit

cosmic shearzs=1

1000

Bκ(ℓ,ℓ,ℓ)

×ℓ3

Bκ(ℓ,ℓ/2,ℓ/2)

×0.2ℓ3

Bκ(ℓ,ℓ,84)

×70ℓ2

Bκ(ℓ,ℓ,510)

×300ℓ2

ℓ


nver

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simulationBiHalofit

cosmic shearzs=1

Fig. 12.— Convergence bispectrum measured from 1000 simulation maps at a source redshift zs = 1 (Sato et al. 2009; Kayo et al. 2013).The black dots are the measurement (the averages and standard deviations), while the red curves are our model prediction. Each map has

a field of view of 5× 5 deg2 and the error bars scale as [(survey are)/(25 deg2))]−1/2. The lower panel plots relative deviations from the redcurves.

though the simulation data have 10% level intrinsic scat-ter. The user can easily incorporate the baryonic effects,calibrated from the IllustrisTNG, in the fitting formula.We also confirm that the formula reproduces the weak-lensing convergence BS measured from light-cone simu-lations.Recently, larger σ8 ≃ 0.81 is inferred from the Planck

results while cosmic shear and galaxy-galaxy lensing pre-fer lower σ8 ≃ 0.77 (the σ8 tension, e.g., MacCrann et al.2015; Abbott et al. 2018; Planck Collaboration 2018a).Combining weak lensing PS and BS can improve the σ8

constraint by a factor of 1.6–3 (e.g., Takada & Jain 2004;Kayo & Takada 2013) and would give a new clue to solvethis issue.

We thank the IllustrisTNG team for making their

simulations publicly available. This work was inpart supported by Grant-in-Aid for Scientific Re-search from the Japan Society for the Promotionof Science (JSPS) (Nos. JP16H03977, JP17H01131,JP17K14273, JP19H00677 and JP19K14767) and MEXTGrant-in-Aid for Scientific Research on InnovativeAreas (No. JP15H05893, JP15H05896, JP15H05899,JP18H04358). KO is supported by JSPS Overseas Re-search Fellowships. TN is supported by Japan Scienceand Technology Agency CREST JPMHCR1414. Numer-ical computations were in part carried out on Cray XC30and XC50 at Centre for Computational Astrophysics,National Astronomical Observatory of Japan.

APPENDIX

A. HALO MODEL

The halo model has been widely used to evaluate the non-linear BS (e.g., Cooray & Hu 2001; Cooray & Sheth 2002;Valageas & Nishimichi 2011; Kayo et al. 2013; Yamamoto et al. 2017). This model assumes that all matter is confinedin halos. The basic halo properties are characterized by the mass function, dn(M)/dM , the spherical density profile,ρ(r;M), and the first- and second-order halo biases, b1,2(M), for a given mass M . In this model, the matter BS isdecomposed into three terms: one- (1h), two- (2h), and three-halo (3h) terms. The 2h term fills the gap between the1h and 3h terms and gives a minor contribution at intermediate scale except for the squeezed limit. By ignoring thisterm, the BS is given as

B(k1, k2, k3) = BHM1h (k1, k2, k3) +BHM

3h (k1, k2, k3). (A1)

The 1h and 3h terms dominate at small and large scales, respectively. The 1h term comes from the density profile ofa single halo:

BHM1h (k1, k2, k3) =

∫

dMdn(M)

dM

(

M

ρ

)3

u(k1;M)u(k2;M)u(k3;M), (A2)


where ρ is the cosmic mean density and u(k;M) is the Fourier transform of the scaled density profile, ρ(r;M)/M . The3h term comes from a spacial correlation among three different halos:

BHM3h (k1, k2, k3) = I11 (k1)I

11 (k2)I

11 (k3)Btree(k1, k2, k3) +

[

I11 (k1)I11 (k2)I

21 (k3)PL(k1)PL(k2) + 2 perm.

]

= 2

[

F2(k1,k2) +I21 (k3)

2I11 (k3)

]

I11 (k1)I11 (k2)I

11 (k3)PL(k1)PL(k2) + 2 perm. (A3)

with

Iβ1 (k) =

∫

dMdn(M)

dM

M

ρbβ(M)u(k;M). (A4)

B. FITTING FORMULA

Our fitting formula adopts the Halofit parameterization for non-linear PS (Smith et al. 2003). The dimensionlesslinear power spectrum is defined as ∆2

L(k) = k3PL(k)/(2π2). The non-linear scale k−1

NL is determined via

σ2(k−1NL) = 1, σ2(R) =

∫

d ln k∆2L(k) e

−k2R2

. (B1)

The effective spectral index at kNL is defined as

neff + 3 = −d lnσ2(R)

d lnR

∣

∣

∣

∣

R=k−1

NL

. (B2)

We also introduce a scaled wavenumber, qi, defined as qi = ki/kNL (i = 1, 2 and 3). Note that these quantities, kNL

and neff , are evaluated for a given redshift. These parameters are identical to those defined in Smith et al. (2003).The fitting function is the sum of 1h and 3h terms:

B(k1, k2, k3) = B1h(k1, k2, k3) +B3h(k1, k2, k3). (B3)

The 1h term is

B1h(k1, k2, k3) =

3∏

i=1

[

1

anqαn

i + bnqβn

i

1

1 + (cnqi)−1

]

. (B4)

Here, we assume that B1h is a product of identical functions of q1, q2 and q3 similar to the halo model, BHM1h , in

Eq. (A2), which is given by a product of u(ki). The 3h term is

B3h(k1, k2, k3) = 2 [F2(k1,k2) + dnq3] I(k1)I(k2)I(k3)PE(k1)PE(k2) + 2 perm., (B5)

with

PE(k) =1 + fnq

2

1 + gnq + hnq2PL(k) +

1

mnqµn + nnqνn1

1 + (pnq)−3 , I(k) =

1

1 + enq. (B6)

Here, PE(k) is an “enhanced” PS which is the linear PS with adding a small-scale enhancement. The first (second)term of PE is similar to the 2h (1h) term in Halofit for the non-linear PS. Similarly, I(k) and dnq correspond to I11 (k)and I21 (k)/[2I

11 (k)], respectively, in the halo model. This 3h term approaches the tree level in the low k limit.

The above fitting parameters (an, bn, ...) are given as polynomials in terms of neff and log10 σ8. Here, σ8 is thespherical overdensity with a radius of 8 h−1Mpc at redshift z (= σ8(z = 0) multplied by the linear growth factor). Theparameters determining the amplitude (i.e., an, bn,mn and nn) are given as functions of log10 σ8, while the most of othersare functions of neff . Since B(k1, k2, k3) and P (k) have the dimensions of [(Length)6] and [(Length)3], respectively,the parameters an and bn have [(Length)−2], mn and nn have [(Length)−3], and the others are dimensionless. Though[Length] is an arbitrary length unit, one may choose [h−1 Mpc] or [Mpc].The fitting parameters of the 1h term are

log10 an = −2.167− 2.944 log10 σ8 − 1.106 (log10 σ8)2 − 2.865 (log10 σ8)

3 − 0.310 rγn

1 ,

log10 bn = −3.428− 2.681 log10 σ8 + 1.624 (log10 σ8)2− 0.095 (log10 σ8)

3,

log10 cn = 0.159− 1.107neff,

log10 αn = min

[

−4.348− 3.006neff − 0.5745n2eff + 10−0.9+0.2neff r22 , 1−

2

3ns

]

,

log10 βn = −1.731− 2.845neff − 1.4995n2eff − 0.2811n3

eff + 0.007 r2,

log10 γn = 0.182 + 0.570neff, (B7)

18 Takahashi et al.

104

105

106

0.1 1 10

B(k,k,k)×k3

B(k,k/2,k/2)×0.2k3

B(k,k,0.045)×k B(k,k,0.45)×10k2

k (h/Mpc)


mat

ter

bisp

ectr

um z=0.55

tree 3h

1h1h+3h

0.1 1 10

B(k,k,k)×k3

B(k,k/2,k/2)×0.2k3

B(k,k,0.045)×k B(k,k,0.45)×10k2

k (h/Mpc)


mat

ter

bisp

ectr

um z=0.55

tree 3h

1h1h+3h

106

107

0.1 1

B(k,k,k)×k3

B(k,k/2,k/2)×0.2k3

B(k,k,0.045)×k B(k,k,0.45)×10k2

k (h/Mpc)


mat

ter

bisp

ectr

um z=0.55

tree 3h

1h1h+3h

1 10

B(k,k,k)×k3

B(k,k/2,k/2)×0.2k3

B(k,k,0.045)×k B(k,k,0.45)×10k2

k (h/Mpc)


mat

ter

bisp

ectr

um z=0.55

tree 3h

1h1h+3h

Fig. 13.— One (1h) and three-halo (3h) term contributions to the total matter bispectrum in the fitting formula.

where r1,2 are ratios of the minimum (kmin), middle (kmid), and maximum (kmax) wavenumbers of the triangle:

r1 =kmin

kmax, r2 =

kmid + kmin − kmax

kmax. (B8)

These r1,2 terms effectively include a “halo triaxiality” in the 1h term (Smith et al. 2006): r1,2 → 0 for the squeezedcase (kmin ≪ kmid ≃ kmax), r1 (r2) → 0.5 (0) for the flattened (kmin ≃ kmid ≃ kmax/2), and r1,2 → 1 for the equilateral

(kmin ≃ kmid ≃ kmax). These terms slightly enhance (suppress) the squeezed (equilateral) BS at k & 5 hMpc−1. Weset the maximum αn value to αn,max = 1− (2/3)ns (where ns is the spectral index of initial PS) so that the 1h termshould be less than the tree-level in the low-k limit.The parameters of the 3h term are

log10 fn = −10.533− 16.838neff − 9.3048n2eff − 1.8263n3

eff,

log10 gn = 2.787 + 2.405neff + 0.4577n2eff,

log10 hn = −1.118− 0.394neff,

log10 mn = −2.605− 2.434 log10 σ8 + 5.710 (log10 σ8)2,

log10 nn = −4.468− 3.080 log10 σ8 + 1.035 (log10 σ8)2,

log10 µn = 15.312 + 22.977neff + 10.9579n2eff + 1.6586n3

eff,

log10 νn = 1.347 + 1.246neff + 0.4525n2eff,

log10 pn = 0.071− 0.433neff,

log10 dn = −0.483 + 0.892 log10 σ8 − 0.086Ωm,

log10 en = −0.632 + 0.646neff. (B9)

Here, Ωm is the matter density parameter at z. Since the calibration is done up to z = 10, the formula should beswitched to the tree-level at z > 10.Figure 13 plots B1h and B3h separately at z = 0.55 for the Planck 2015. Here, these are unbinned results. As clearly

seen, for the equilateral and flattened cases, the 1h (3h) term dominates at small (large) scale. For the squeezed case,the 1h (3h) term dominates if k3 is in the non-linear (linear) regime.Figure 14 shows a binning effect on BS. As shown clearly, the squeezed BS is sensitive to the binning. This is because

the cosine term in the F2 kernel is very sensitive to the squeezed triangle configuration (for detailed discussion, seesection IIB of Namikawa et al. 2019).

C. FITTING TO THE RATIO OF BISPECTRUM WITH TO WITHOUT BARYONS

In this Appendix, we present a fitting function of ratio of BS with to without baryons, Rb defined in Eq. (13),calibrated with the TNG300-1 simulation (Nelson et al. 2019). We include the triangle configurations (k1, k2, k3)satisfying the following two conditions: a) the number of triangles in the bin is larger than 106 in order to removenoisy data points and b) the shot noise contribution is less than 3%. This fitting range is from k = 0.03 to 100 hMpc−1

at z = 0–5.We fit the BS ratio Rb as

Rb(k1, k2, k3) =3∏

i=1

[

A0 exp

−

∣

∣

∣

∣

xi − µ0

σ0

∣

∣

∣

∣

α0

−A1 exp

−

(

xi − µ1

σ1

)2

+

(

kik∗

)α2

+ 1

β2

]

, (C1)


104

105

106

0.1 1 10

B(k,k,k)×k3

B(k,k/2,k/2)×0.2k3

B(k,k,0.045)×k B(k,k,0.45)×10k2

k (h/Mpc)


mat

ter

bisp

ectr

um z=0.55

unbinnedbinned ∆logk=0.1

0.1 1 10

B(k,k,k)×k3

B(k,k/2,k/2)×0.2k3

B(k,k,0.045)×k B(k,k,0.45)×10k2

k (h/Mpc)


mat

ter

bisp

ectr

um z=0.55


106

107

0.1 1

B(k,k,k)×k3

B(k,k/2,k/2)×0.2k3

B(k,k,0.045)×k B(k,k,0.45)×10k2

k (h/Mpc)


mat

ter

bisp

ectr

um z=0.55


1 10

B(k,k,k)×k3

B(k,k/2,k/2)×0.2k3

B(k,k,0.045)×k B(k,k,0.45)×10k2

k (h/Mpc)


mat

ter

bisp

ectr

um z=0.55


Fig. 14.— Effect of k−binning on the bispectrum fitting formula. The black curve is the unbinned result, while the dashed red curve isthe binned one with bin-width ∆ log10 k = 0.1.

where xi = log10[ki/(hMpc−1)]. These fitting parameters are given in terms of the scale factor a:

A0 = 0.068 (a− 0.5)0.47

Θ(a− 0.5),

µ0 = 0.018 a+ 0.837 a2,

σ0 = 0.881µ0,

α0 = 2.346,

A1 = 1.052 (a− 0.2)1.41

Θ(a− 0.2),

µ1 =∣

∣0.172 + 3.048 a− 0.675 a2∣

∣ ,

σ1 = (0.494− 0.039 a)µ1,

k∗ = 29.90− 38.73 a+ 24.30 a2,

α2 = 2.25,

α2β2 =0.563

(a/0.06)0.02

+ 1, (C2)

where k∗ has the unit of hMpc−1 and Θ(x) is the step function; Θ(x) = 1 and 0 for x ≥ 0 and x < 0, respectively. Thefirst term of Eq. (C1) represents the small enhancement at intermediate scale (k ≃ 1–10 hMpc−1) at low z (< 1), thesecond term is the depression at k ≈ 1 hMpc−1, and the last term is the strong enhancement at high k (& 10 hMpc−1).The ratio Rb approaches unity in the low-k limit.

REFERENCES

Abbott, T. M. C., Abdalla, F. B., Alarcon, A., et al. 2018,Phys. Rev. D, 98, 043526

Angulo, R. E., Foreman, S., Schmittfull, M., & Senatore, L. 2015,J. Cosmol. Astropart. Phys. , 2015, 039

Bartelmann, M., & Schneider, P. 2001, Phys. Rep., 340, 291Beck, D., Fabbian, G., & Errard, J. 2018, Phys. Rev. D, 98,

043512Berge, J., Amara, A., & Refregier, A. 2010, The Astrophysical

Journal, 712, 992Bernardeau, F., Colombi, S., Gaztanaga, E., & Scoccimarro, R.

2002a, Phys. Rep., 367, 1Bernardeau, F., Mellier, Y., & van Waerbeke, L. 2002b, A&A,

389, L28Bohm, V., Schmittfull, M., & Sherwin, B. D. 2016, Phys. Rev. D,

94, 043519Bose, B., Byun, J., Lacasa, F., Moradinezhad Dizgah, A., &

Lombriser, L. 2019, arXiv e-prints, arXiv:1909.02504Bose, B., & Taruya, A. 2018, J. Cosmol. Astropart. Phys. , 2018,

019Byun, J., Eggemeier, A., Regan, D., Seery, D., & Smith, R. E.

2017, MNRAS, 471, 1581Chan, K. C., & Blot, L. 2017, Phys. Rev. D, 96, 023528Chan, K. C., Moradinezhad Dizgah, A., & Norena, J. 2018,

Phys. Rev. D, 97, 043532

Chisari, N. E., Richardson, M. L. A., Devriendt, J., et al. 2018,MNRAS, 480, 3962

Chisari, N. E., Mead, A. J., Joudaki, S., et al. 2019, The OpenJournal of Astrophysics, 2, 4

Colavincenzo, M., Sefusatti, E., Monaco, P., et al. 2019, MNRAS,482, 4883

Cooray, A., & Hu, W. 2001, The Astrophysical Journal, 548, 7Cooray, A., & Sheth, R. 2002, Phys. Rep., 372, 1Coulton, W. R., Liu, J., Madhavacheril, M. S., Bohm, V., &

Spergel, D. N. 2019, Journal of Cosmology and Astro-ParticlePhysics, 2019, 043

Crocce, M., Pueblas, S., & Scoccimarro, R. 2006, MNRAS, 373,369

DeRose, J., Wechsler, R. H., Tinker, J. L., et al. 2019, ApJ, 875,69

Fabbian, G., Lewis, A., & Beck, D. 2019, arXiv e-prints,arXiv:1906.08760

Foreman, S., Coulton, W., Villaescusa-Navarro, F., & Barreira, A.2019, arXiv e-prints, arXiv:1910.03597

Fu, L., Kilbinger, M., Erben, T., et al. 2014, MNRAS, 441, 2725Garrison, L. H., Eisenstein, D. J., Ferrer, D., et al. 2018, ApJS,

236, 43Gatti, M., Chang, C., Friedrich, O., et al. 2019, arXiv e-prints,

arXiv:1911.05568

20 Takahashi et al.

Gil-Marın, H., Wagner, C., Fragkoudi, F., Jimenez, R., & Verde,L. 2012, J. Cosmol. Astropart. Phys. , 2, 047

Gorski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759Hamana, T., Shirasaki, M., Miyazaki, S., et al. 2019, arXiv

e-prints, arXiv:1906.06041Hanson, D., Rocha, G., & Gorski, K. 2009, MNRAS, 400, 2169Harnois-Deraps, J., van Waerbeke, L., Viola, M., & Heymans, C.

2015, MNRAS, 450, 1212Hartlap, J., Simon, P., & Schneider, P. 2007, A&A, 464, 399Hashimoto, I., Rasera, Y., & Taruya, A. 2017, Phys. Rev. D, 96,

043526Hearin, A. P., Zentner, A. R., & Ma, Z. 2012, J. Cosmol.

Astropart. Phys. , 4, 034Heitmann, K., Higdon, D., White, M., et al. 2009, ApJ, 705, 156Hellwing, W. A., Schaller, M., Frenk, C. S., et al. 2016, MNRAS,

461, L11Hikage, C., Oguri, M., Hamana, T., et al. 2019, PASJ, 71, 43Hinshaw, G., Larson, D., Komatsu, E., et al. 2013, ApJS, 208, 19Hu, W., & Okamoto, T. 2002, ApJ, 574, 566Huterer, D., & Takada, M. 2005, Astroparticle Physics, 23, 369Jarvis, M., Bernstein, G., & Jain, B. 2004, MNRAS, 352, 338Jenkins, A., Frenk, C. S., Pearce, F. R., et al. 1998, ApJ, 499, 20Jing, Y. P. 2005, ApJ, 620, 559Joachimi, B., Cacciato, M., Kitching, T. D., et al. 2015,

Space Sci. Rev., 193, 1Kayo, I., & Takada, M. 2013, arXiv e-prints, arXiv:1306.4684Kayo, I., Takada, M., & Jain, B. 2013, MNRAS, 429, 344Kilbinger, M., & Schneider, P. 2005, A&A, 442, 69Kilbinger, M., Heymans, C., Asgari, M., et al. 2017, MNRAS,

472, 2126Kitching, T. D., Alsing, J., Heavens, A. F., et al. 2017, MNRAS,

469, 2737Knabenhans, M., Stadel, J., Marelli, S., et al. 2019, MNRAS, 484,

5509Lawrence, E., Heitmann, K., Kwan, J., et al. 2017, ApJ, 847, 50Lazanu, A., Giannantonio, T., Schmittfull, M., & Shellard,

E. P. S. 2016, Physical Review D, 93, 083517Lazanu, A., & Liguori, M. 2018, J. Cosmol. Astropart. Phys. ,

2018, 055Lewis, A., & Challinor, A. 2006, Phys. Rep., 429, 1Lewis, A., Challinor, A., & Lasenby, A. 2000, ApJ, 538, 473Lewis, A., & Pratten, G. 2016, J. Cosmol. Astropart. Phys. ,

2016, 003MacCrann, N., Zuntz, J., Bridle, S., Jain, B., & Becker, M. R.

2015, MNRAS, 451, 2877Madhavacheril, M. S., & Hill, J. C. 2018, Phys. Rev., D98, 023534Marozzi, G., Fanizza, G., Di Dio, E., & Durrer, R. 2018,

Phys. Rev. D, 98, 023535McCullagh, N., Jeong, D., & Szalay, A. S. 2016, MNRAS, 455,

2945Mead, A. J., Peacock, J. A., Heymans, C., Joudaki, S., &

Heavens, A. F. 2015, MNRAS, 454, 1958Mishra, N., & Schaan, E. 2019, arXiv:1908.08057Munshi, D., Namikawa, T., Kitching, T. D., et al. 2019, arXiv

e-prints, arXiv:1910.04627Munshi, D., Smidt, J., Heavens, A., Coles, P., & Cooray, A. 2011,

MNRAS, 411, 2241Namikawa, T. 2016, Phys. Rev. D, 93, 121301Namikawa, T., Bose, B., Bouchet, F. R., Takahashi, R., &

Taruya, A. 2019, Phys. Rev. D, 99, 063511Namikawa, T., Hanson, D., & Takahashi, R. 2013, MNRAS, 431,

609Nelson, D., Springel, V., Pillepich, A., et al. 2019, Computational

Astrophysics and Cosmology, 6, 2Nishimichi, T., Shirata, A., Taruya, A., et al. 2009, PASJ, 61, 321Nishimichi, T., Takada, M., Takahashi, R., et al. 2019, ApJ, 884,

29Osato, K., Shirasaki, M., & Yoshida, N. 2015, ApJ, 806, 186Osborne, S. J., Hanson, D., & Dore, O. 2014, J. Cosmol.

Astropart. Phys. , 03, 024Planck Collaboration. 2016, A&A, 594, A13—. 2018a, arXiv e-prints, arXiv:1807.06209

—. 2018b, arXiv e-prints, arXiv:1807.06210—. 2019, arXiv e-prints, arXiv:1905.05697Pratten, G., & Lewis, A. 2016, J. Cosmol. Astropart. Phys. ,

2016, 047

Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery,B. P. 2002, Numerical recipes in C++ : the art of scientificcomputing

Rampf, C., & Wong, Y. Y. Y. 2012, J. Cosmol. Astropart. Phys. ,2012, 018

Rizzato, M., Benabed, K., Bernardeau, F., & Lacasa, F. 2018,arXiv e-prints, arXiv:1812.07437

Sato, M., Hamana, T., Takahashi, R., et al. 2009, ApJ, 701, 945Sato, M., & Nishimichi, T. 2013, Phys. Rev. D, 87, 123538Schneider, A., Teyssier, R., Potter, D., et al. 2016, J. Cosmol.

Astropart. Phys. , 4, 047Scoccimarro, R. 1997, ApJ, 487, 1—. 2015, Phys. Rev. D, 92, 083532Scoccimarro, R., Colombi, S., Fry, J. N., et al. 1998, ApJ, 496, 586Scoccimarro, R., & Couchman, H. M. P. 2001, MNRAS, 325, 1312Scoccimarro, R., & Frieman, J. A. 1999, ApJ, 520, 35Sefusatti, E., Crocce, M., & Desjacques, V. 2010, MNRAS, 406,

1014Sefusatti, E., Crocce, M., Pueblas, S., & Scoccimarro, R. 2006,

Phys. Rev. D, 74, 023522Sefusatti, E., Crocce, M., Scoccimarro, R., & Couchman,

H. M. P. 2016, MNRAS, 460, 3624Semboloni, E., Heymans, C., van Waerbeke, L., & Schneider, P.

2008, MNRAS, 388, 991Semboloni, E., Hoekstra, H., & Schaye, J. 2013, MNRAS, 434, 148Semboloni, E., Hoekstra, H., Schaye, J., van Daalen, M. P., &

McCarthy, I. G. 2011a, MNRAS, 417, 2020Semboloni, E., Schrabback, T., van Waerbeke, L., et al. 2011b,

MNRAS, 410, 143Shi, X., Joachimi, B., & Schneider, P. 2010, A&A, 523, A60Shirasaki, M., Hamana, T., & Yoshida, N. 2015, MNRAS, 453,

3043Simon, P., Semboloni, E., van Waerbeke, L., et al. 2015, MNRAS,

449, 1505Smith, R. E., Watts, P. I. R., & Sheth, R. K. 2006, MNRAS, 365,

214Smith, R. E., Peacock, J. A., Jenkins, A., et al. 2003, MNRAS,

341, 1311Spergel, D. N., Bean, R., Dore, O., et al. 2007, ApJS, 170, 377Springel, V. 2005, MNRAS, 364, 1105Springel, V., Yoshida, N., & White, S. D. M. 2001, New Astron.,

6, 79Springel, V., Pakmor, R., Pillepich, A., et al. 2018, MNRAS, 475,

676Sugiyama, N. S., Saito, S., Beutler, F., & Seo, H.-J. 2019, arXiv

e-prints, arXiv:1908.06234Takada, M., & Hu, W. 2013, Phys. Rev. D, 87, 123504Takada, M., & Jain, B. 2002, MNRAS, 337, 875—. 2004, MNRAS, 348, 897Takahashi, R., Hamana, T., Shirasaki, M., et al. 2017, ApJ, 850,

24Takahashi, R., Sato, M., Nishimichi, T., Taruya, A., & Oguri, M.

2012, ApJ, 761, 152Troxel, M. A., & Ishak, M. 2012, MNRAS, 419, 1804—. 2015, Phys. Rep., 558, 1Valageas, P., & Nishimichi, T. 2011, A&A, 532, A4Valageas, P., Sato, M., & Nishimichi, T. 2012, A&A, 541, A161van Daalen, M. P., Schaye, J., Booth, C. M., & Dalla Vecchia, C.

2011, MNRAS, 415, 3649van Engelen, A., Bhattacharya, S., Sehgal, N., et al. 2014, ApJ,

786, 14van Uitert, E., Joachimi, B., Joudaki, S., et al. 2018, MNRAS,

476, 4662Van Waerbeke, L., Benjamin, J., Erben, T., et al. 2013, MNRAS,

433, 3373Yamamoto, K., Nan, Y., & Hikage, C. 2017, Phys. Rev. D, 95,

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