A semi-analytic model of pairwise velocity distribution between dark matter halos
Masato Shirasaki, Eric M. Huff, Katarina Markovic, Jason D. Rhodes
DDraft version August 10, 2020
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A semi-analytic model of pairwise velocity distribution between dark matter halos
Masato Shirasaki,
1, 2, 3
Eric M. Huff, Katarina Markovic, and Jason D. Rhodes
2, 4 National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA, USA Overseas Research Fellowships by the Japan Society for the Promotion of Science (JSPS) Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study(UTIAS), The University of Tokyo, Chiba 277-8583, Japan
Submitted to ApJABSTRACTWe study the probability distribution function (PDF) of relative velocity between two different darkmatter halos (i.e. pairwise velocity) with a set of high-resolution cosmological N -body simulations. Werevisit a non-Gaussian framework to predict pairwise velocity statistics developed in Tinker (2007) andcalibrate a statistical relationship between the pairwise velocity and an environmental mass densityaround halos in the framework. We investigate the pairwise velocity PDFs over a wide range ofhalo masses of 10 . < ∼ M [ h − M (cid:12) ] < ∼ and redshifts of 0 < z <
1. At a given set of masses,redshift and the separation length between two halos, our model requires three parameters to set thepairwise velocity PDF, whereas previous non-Gaussian models in the literature assume four or more freeparameters. At the length scales of 5 < r [ h − Mpc] <
40, our model predicts the mean and dispersionof the pairwise velocity for dark matter halos with their masses of 10 . < ∼ M [ h − M (cid:12) ] < ∼ . at0 . < z < O (10) h − Mpc between redshift and real space for a given real-spacecorrelation function. For a mass-limited halo sample with their masses greater than 10 . h − M (cid:12) at z = 0 .
55, our model can explain the monopole and quadropole moments of the redshift-space two-pointcorrelations with a precision better than 5% at the scales of 5 −
40 and 10 − h − Mpc, respectively.Our model of the pairwise velocity PDF will give a detailed explanation of statistics of massive galaxiesat the intermediate scales in redshift surveys, including the non-linear redshift-space distortion effectin two-point correlation functions and the measurements of the kinematic Sunyaev-Zel’dovich effect.
Keywords: cosmology: large-scale structure of universe — galaxies: halos — methods: numerical INTRODUCTIONAccelerating expansion of the late-time Universe is a long-standing mystery in modern astronomy (e.g. Weinberget al. 2013, for a review about observational probes). There are two leading physical models to solve the most surprisingcosmological discovery in many decades. One is the dark energy model which assumes an exotic form of energy inthe Universe, and the other requires a modification of General Relativity at long-length and weak-force regimes. Todistinguish the two models in an observational way, one needs detailed measurements of cosmic mass density andvelocity over a large volume in the Universe. A modification of gravity can induce the scale dependence on thegravitational growth and flow of cosmic mass density even at linear scales, while the large-scale density growth and
Corresponding author: Masato [email protected] a r X i v : . [ a s t r o - ph . C O ] A ug Shirasaki et al. flow can be uniquely determined by the expansion rate of the Universe alone in the presence of a smooth uniform darkenergy under General Relativity (but see, e.g. Jain & Zhang 2008, for more detailed discussions).Among observational probes, redshift surveys of distant galaxies are one of the most promising approaches toinvestigating density and velocity fields. In the standard theory of formation of large-scale structures, galaxies arethought to be a tracer of underlying cosmic mass density. Their clustering properties contain rich cosmologicalinformation in principle. The main challenges to use galaxies for cosmological studies are that galaxies are a biasedtracer of density fields and a galaxy bias may depend on various factors. Throughout this paper, we consider a halo-based model for the galaxy bias. Dark matter halos are gravitationally-bounded objects formed in cosmic matterdistributions and a building block of large-scale structures (see, Cooray & Sheth 2002, for a review of the halo model).In the halo model, one commonly assumes that dark matter halos host some galaxies and the number of galaxies insingle halos depends on the halo mass alone. These assumptions enable us to explain the observed clustering of avariety of galaxies in a precise way (e.g. Zehavi et al. 2005; Cooray 2006; Zehavi et al. 2011). Nevertheless, the commonhalo-galaxy connection may cause serious systematic errors in galaxy-based cosmological analyses if the galaxy biasdepends on other properties (e.g. Croton et al. 2007; Zentner et al. 2014).The two-point correlation function ξ gg is a common observable in galaxy redshift surveys to measure the clusteringof galaxies. In a universe with statistical isotropy, the two-point correlation depends on the separation length betweengalaxies alone. However, actual surveys rely on the observation of redshifts to infer the line-of-sight distances toindividual galaxies. A redshift z in the spectrum of each galaxy is caused by cosmic expansion as well as the peculiarvelocity of the galaxy itself. Hence, the spatial coordinate inferred by the observation of redshifts is different from thetrue counterpart. This effect is known as redshift-space distortions and the observed spatial separation (i.e. in redshiftspace) for a given pair of galaxies is then expressed as s p = r p , (1) s π = r π + (1 + z ) v z H ( z ) , (2)where H ( z ) is the Hubble parameter at z , r p and r π represents the perpendicular and parallel components with respectto the line-of-sight direction, s p and s π are the counterparts in redshift space, and v z is the relative velocity betweentwo galaxies (i.e. the pairwise velocity) along the line of sight. In redshift space, the line of sight toward each galaxyis a special direction and the observed two point correlation function depends on both of s p and s π . The relation ofthe two-point correlation between real and redshift space is given by (e.g. Peebles 1980; Scoccimarro 2004),1 + ξ S gg ( s p , s π ) = (cid:90) ∞−∞ H ( z ) d r π (1 + z ) P g (cid:18) v z = H ( z )( s π − r π )(1 + z ) (cid:12)(cid:12)(cid:12) s p , r π (cid:19) (cid:104) ξ gg ( (cid:113) s p + r π ) (cid:105) , (3)where P g is the probability distribution function (PDF) of the line-of-sight pairwise velocity of galaxies, and ξ S gg isthe two-point correlation function in redshift space. Therefore, it is essential to develop an accurate theoretical modelof the pairwise velocity statistics as well as the real-space correlation function for cosmological analyses with redshiftsurveys.Measurements of small-scale streaming motions of galaxies may bring meaningful information to improve our under-standing of the halo-galaxy relationship. Xu & Zheng (2018) have shown that the dispersion of the pairwise velocitybetween two dark matter halos depends on properties other than the halo mass, while a similar effect has been foundin a semi-analytic galaxy formation model (Padilla et al. 2019). Apart from the halo-galaxy relationship, numericalsimulations have shown that a modification of gravity can change the streaming motion between two massive-galaxy-sized halos with a separation length of ∼ − airwise velocity between dark matter halos N -body simulations, we calibrate a physically-intuitive and efficient model of the pairwise velocity developed by Tinker(2007) over a range of halo masses 10 . < M [ h − M (cid:12) ] < at 0 < z <
1. We then validate that our model canreproduce the two-point correlation function in redshift space including the non-linear distortion effects due to thepeculiar velocity of galaxies. We also study information contents of the pairwise velocity statistics in galaxy clusteringanalyses.The paper is organized as follows. In Section 2, we present an overview of the Tinker model and introduce ourrevised model. In Section 3, we describe the N -body simulations, mock galaxy catalogs, and clustering statistics usedin this paper. We summarize our calibration process of the model parameters based on three non-zero moments in thepairwise velocity in Section 4. The results are presented in Section 5 and we mention the limitations in our model inSection 6. Finally, the conclusions and discussions are provided in Section 7. PAIRWISE VELOCITY DISTRIBUTION OF DARK MATTER HALOSIn this section, we briefly introduce a theoretical framework to predict the pairwise velocity statistics of dark matterhalos proposed in Tinker (2007). We then present our new model with some modifications. Table 1 summarizes themodel parameters in the framework and those are dependent on halo masses, redshifts, and separation lengths.2.1.
Setup
Consider a pair of two halos with their masses of M and M at a given redshift z . The pairwise velocity for thehalo pair is then defined as the relative velocity between the two halos, v ( r ) ≡ v ( r ) − v ( r ) , (4)where v i and r i represent the velocity and position of the i -th halo ( i = 1 , r ≡ r − r . Throughout thispaper, we define the position of a given halo in the comoving coordinate, while the velocity is defined in the physicalcoordinate. Assuming a spherically symmetric phase-space distribution of halos, we simplify v ( r ) = v ( r ) where r = | r | . In this paper, we seek a physically-motivated and efficient model of the probability distribution function (PDF)of the pairwise velocity. In particular, we study the dependence of the pairwise velocity PDF on halo masses ( M and M ), redshifts z and the separation length r . Since the line-of-sight component of the pairwise velocity is relevant tostatistical analyses in redshift surveys in practice (also see Section 3.3), we focus on two different components in thethree-dimensional velocity v . One is the radial component of v r and another is the half of the tangential components v t at each radius r . We define these two components as v r ≡ v · r /r, (5) v t ≡ v x cos θ cos φ + v y cos θ sin φ − v z sin θ, (6)where we set a Cartesian coordinate system of r /r = (sin θ cos φ, sin θ sin φ, cos θ ), the angle θ is defined by theopening angle between the line-of-sight direction and the vector r , v x represents the x -axis component of v in theCartesian coordinate and so on. Note that v z corresponds to the line-of-sight velocity in this notation and it holds v z = v r cos θ − v t sin θ . 2.2. The Tinker model
Tinker (2007) proposed an analytic model of the joint PDF of P ( v r , v t | r, M , M , z ) for a pair of dark matter halosby assuming the following conditions:(i) There exists a latent variable to make the joint PDF non-Gaussian, but the PDF at a given latent variable canbe approximated as Gaussian.(ii) At a given latent variable, v r and v t are assumed to be independent.(iii) The latent variable for modeling of P ( v r , v t | r, M , M , z ) is set to the local environmental mass overdensity δ around a pair of halos. Shirasaki et al.
These conditions allow us to express the joint PDF as the following functional form: P ( v r , v t | r, M , M , z ) = (cid:90) d δ N ( v t | µ t ( δ ) , Σ t ( δ )) N ( v r | µ r ( δ ) , Σ r ( δ )) F ( δ | r, M , M , z ) , (7)where N ( u | µ, Σ) represents a Gaussian PDF of a one-dimensional random field u with the mean of µ and the varianceof Σ and F ( δ | r, M , M , z ) is the conditional PDF of the mass over-density field δ when one finds the halo pair withtheir masses of M and M at the redshift z within the radius of r . In Eq. (7), Tinker (2007) sets the mean v t at agiven δ to be zero ( µ t = 0), while the variables of Σ t , µ r , and Σ r can depend on the halo masses, redshift and radiusas well as the environmental density δ . In the following, we summarize key ingredients in Eq. (7).2.2.1. Conditional PDF of mass overdensity
The unconditional PDF of smoothed mass density at a smoothing scale of r can be described by a log-normaldistribution (e.g. Coles & Jones 1991; Kofman et al. 1994; Kayo et al. 2001), while Tinker (2007) found the conditionalPDF F in numerical simulations is well fitted by F ( δ | r, M , M , z ) = A exp (cid:20) − ˜ ρ ( r, M , M , z )1 + δ (cid:21) P ln ( δ | r ) , (8)where A is a normalization constant so that (cid:82) d δ F = 1, ˜ ρ is a density cutoff scale to be calibrated with N -bodysimulations, and P ln is the log-normal distribution given by P ln ( δ | r ) = 1 √ πσ ln exp (cid:20) − { ln(1 + δ ) + σ / } σ (cid:21)
11 + δ . (9)In Eq. (9), the variance σ is set to be σ ( r, z ) = ln(1 + σ ( r, z )) where σ ( r, z ) is the non-linear mass variancesmoothed by a top-hat filter at the scale of r at the redshift z . To be specific, the top-hat mass variance is given by σ ( r, z ) = (cid:90) ∞ πk d k (2 π ) W ( kr ) P NL ( k, z ) , (10) W TH ( x ) = 3 (sin x − x cos x ) x , (11)where P NL ( k, z ) is the non-linear matter power spectrum at z . Tinker (2007) assumed that the density cutoff scale ˜ ρ takes the form ˜ ρ ( r, M , M , z ) = ˜ ρ ( b L ( M , z ) + b L ( M , z )) + (cid:18) rr (cid:19) α , (12)where b L ( M, z ) is the linear halo bias at the redshift z , and three parameters of ˜ ρ , r , and α have been calibratedwith a set of N -body simulations. Tinker (2007) found the simulation results can be explained by the form of Eq. (8)when ˜ ρ = 1 . α = − .
2, and r = 9 . × MAX( R , , R , ) where R ,i represents a spherical over-densityradius of the halo of M i . We here define the halo mass by M ≡ M = (4 π/
3) 200¯ ρ m0 R , where ¯ ρ m0 is the meancosmic mass density today and the halo radius R is defined in the comoving coordinate.Throughout this paper, we adopt the linear halo bias model in Tinker et al. (2010), while we compute the non-linearmatter power spectrum by using the fitting formula calibrated by a set of N -body simulations (Takahashi et al. 2012).Note that we use the linear matter power spectrum without the baryon acoustic oscillations (Eisenstein & Hu 1998)when computing σ to avoid any oscillations in the predicted velocity moments at large scales of > ∼ h − Mpc.2.2.2.
Mean and variance at a given environmental density
For a given halo pair at the separation of r and the environmental density δ , Tinker (2007) developed a model of themean infall velocity µ r ( δ, r ) by combining linear theory and the spherical collapse model. The linear theory predictsthe relation of the velocity and overdensity fields as µ lin ( δ, r, z ) = − H ( z )1 + z r f ( z ) δ , (13) airwise velocity between dark matter halos H ( z ) is the Hubble parameter at z , f ( z ) = d ln D/ d ln(1 + z ) − where D ( z ) is the linear growth factor at z .At non-linear scales, the spherical collapse model can provide a reasonable approximation of the mean infall velocity.In the Einstein-de Sitter universe, one can derive the relation of the velocity and density perturbations as µ sc ( δ, r, z ) = H ( z )1 + z r f ( z ) G ( δ ) , (14)where G ( δ ) is expressed in a parametric form as δ = 92 ( γ − sin γ ) (1 − cos γ ) − , G = 32 sin γ ( γ − sin γ )(1 − cos γ ) − . (15)Tinker (2007) then proposed a model by combining Eqs. (13) and (14): µ r ( δ, r, M , M , z ) = w ( r ) µ sc ( δ, r, z ) exp (cid:20) − (cid:16) . r (1+ δ ) (cid:17) (cid:21) + [1 − w ( r )] µ lin ( δ, r, z ) ( r > R cut ) µ sc ( δ cut , r, z ) exp (cid:20) − (cid:16) . r (1+ δ cut ) (cid:17) (cid:21) ( r ≤ R cut ) , (16)where R cut = MAX( R , , R , ), 1 + δ cut = exp( − σ / w ( r ) and the exponential cutoffhave been calibrated against the numerical simulations. Tinker (2007) found that the following weight function showsa reasonable fit to the simulation results, w ( r ) = r [ h − Mpc] ≤ . − .
62 ln r (4 < r [ h − Mpc] ≤ < r [ h − Mpc]) . (17)For the velocity dispersions Σ t,r , Tinker (2007) introduced the following parametric form ofΣ t,r = 200 [km / s] (cid:18) Ω m ( z )0 . (cid:19) . (cid:18) D ( z ) σ . (cid:19) (cid:18) δ ˜ ρ t,r (cid:19) β , (18)where Ω m ( z ) = Ω m0 (1 + z ) / [Ω m0 (1 + z ) + (1 − Ω m0 )] (Ω m0 is the mass-density parameter today), σ is the massvariance for the linear overdensity field at z = 0 when smoothed by the top-hat filter at 8 h − Mpc, and three parameters˜ ρ t,r and β have been calibrated with the simulation results as a function of r , M and M . Note that the scaling withΩ m ( z ) and σ in Eq. (18) is motivated by the linear theory (recall f (cid:39) Ω . ( z )). For M ≥ M , the fitting formulasare summarized as β ( r ) = (cid:18) r h − Mpc (cid:19) . , (19)˜ ρ t ( r, M , M ) = (cid:32) r . R / , (cid:33) − . + (cid:32) r . R / , (cid:33) − . + 0 . , (20)˜ ρ r ( r, M , M ) = (cid:32) r . R / , (cid:33) − . + (cid:32) r . R / , (cid:33) − . + 0 . , (21)where R , = R , + R , in comoving h − Mpc.2.3.
New model
The model by Tinker (2007) is physically-intuitive and efficient to compute the pairwise velocity PDF for dark matterhalos, but we find that it does not provide a reasonable fit to the latest high-resolution simulation results as shown We normalize D ( z ) = 1 at z = 0 throughout this paper. Equation (17) in Tinker (2007) misses the factor of 1 / G . Shirasaki et al.
Table 1.
A short summary of parameters in the model of pairwise velocity PDF of dark matter halos.Model parameters Tinker (2007) This paper Reference˜ ρ Eq. (12) Eq. (22) Density cutoff scale on the halo formation as in Eq. (8) µ r Eq. (16) Eq. (23) Mean radial velocity at a given environmental density˜ ρ t,r Eqs. (20) & (21) Eq. (24) Scale density on the dispersion-density relation as in Eq. (18) in Section 5. There may be several reasons why the model can not reproduce the simulation results today. A majorconcern about the model of Tinker (2007) is that its parameter calibration relies on the results of N -body simulationsin a ΛCDM cosmology with the spectral index n s = 1 and a larger amplitude of the initial density power spectrum at k = 0 .
05 Mpc − than the inferred value from Planck (Planck Collaboration et al. 2016). This can affect the kinematicsof dark matter halos even at large scales, because the linear velocity in Fourier space scales with δ/k where k is thewave number. In addition, the simulations in Tinker (2007) assume Ω m0 = 0 . σ = 0 .
95 at z = 0, which mayresult in sizable differences in the non-linear evolution of cosmic mass density. Furthermore, the simulations consist of360 particles in a volume of 253 [ h − Mpc] and the mass resolution may be less sufficient to study the halo-galaxyconnection in a modern manner. In fact, recent observations of massive galaxies in the Sloan Sky Digital Sky SurveyIII (SDSS III) have shown that the kinematics of galaxies closely relate to the phase-space density in the inner regionsof their host dark matter halos (Reid et al. 2014; Guo et al. 2015), while the halo velocity in Tinker (2007) is definedby the center-of-mass velocity. Detailed simulations show that halo cores are not at rest relative to the halo bulk(Behroozi et al. 2013). High-resolution and large-volume cosmological simulations would be needed to re-calibrate themodel of Tinker (2007) and this is the scope of this paper.Our new model follows the basic concept in Tinker (2007), but we introduce minor revisions so that the final modelcan reproduce the latest simulation results over a wide range of halo masses, redshifts, and separation lengths. For theconditional PDF of finding a halo pair given a mass density, we adopt the exponential cutoff as in Eq. (8) to effectivelyinclude the environmental dependence of halo formation and parametrize the density cutoff scale as in Eq. (12), butwe allow a more complicated mass and redshift dependence:˜ ρ ( r, M , M , z ) = A ρ ( M , M , z ) ( b L ( M , z ) + b L ( M , z )) + B ρ ( M , M , z ) (cid:18) r MAX( R , , R , ) (cid:19) C ρ ( M ,M ,z ) , (22)Also, we modify the functional form of mean radial velocity at a given overdensity, µ r ( δ ), as µ r ( δ, r, z ) = − H ( z )1 + z r f ( z )3 δ c (cid:104) (1 + δ ) /δ c − (cid:105) , (23)where δ c = 1 . δ → ρ t,r onhalo masses and radius by using a double power-law form of˜ ρ t,r ( r, M , M , z ) = C (0) t,r ( M , M , z ) r p t,r ( M ,M ,z ) + C (1) t,r ( M , M , z ) r q t,r ( M ,M ,z ) + C (2) t,r ( M , M , z ) , (24)where we introduce five functions of C ( i ) ( i = 0 − p and q for each velocity dispersion. Note that the power-lawindex of Eq. (18) is fixed to Eq. (19) in the new model as well.The detailed forms of A ρ , B ρ , C ρ , C ( i ) t,r ( i = 0 − p t,r , and q t,r are found in Appendix A. We also provide the detailsof our calibration process to find the forms of various functions in Section 4. DATA3.1. N -body simulations and halo catalogs airwise velocity between dark matter halos Table 2.
The number of dark matter halos analyzed in this paper. Note that the halo mass is defined by the mass of a sphericaloverdensity, with 200-times the mean density of the universe.Halo mass z = 0 z = 0 . z = 0 . z = 1 . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < To study the pairwise velocity statistics of dark matter halos, we use a set of publicly available halo catalogsprovided by the ν GC collaboration . Ishiyama et al. (2015) performed a series of high-resolution cosmological (dark-matter-only) N -body simulations with various combinations of mass resolutions and volumes on the basis of theΛCDM cosmology consistent with observational results obtained by the Planck satellite. Among them, we use thehalo catalogs based on the largest-volume run called ν GC-L run, which consists of 8192 dark matter particlesin a box of 1 . h − Gpc. The corresponding mass resolution is 2 . × h − M (cid:12) , allowing us to study the corevelocity of dark matter halos in a robust way. The simulations were performed by a massive parallel TreePM code ofGreeM (Ishiyama et al. 2009, 2012) on the K computer at the RIKEN Advanced Institute for Computational Science,and Aterui super-computer at Center for Computational Astrophysics (CfCA) of National Astronomical Observatoryof Japan. The authors generated the initial conditions by a publicly available code, 2LPTic , using second-orderLagrangian perturbation theory (e.g. Crocce et al. 2006), as well as the online version of CAMB (Lewis et al. 2000)to set the linear power spectrum at the initial redshift of z = 127. In the simulations, the following cosmologicalparameters were adopted: Ω m0 = 0 .
31, Ω b0 = 0 . Λ = 1 − Ω m0 = 0 . h = 0 . n s = 0 .
96, and σ = 0 .
83. Theseare consistent with Planck (Planck Collaboration et al. 2016).In this paper, we use the halo catalogs produced with the ROCKSTAR halo finder (Behroozi et al. 2013) at four differ-ent redshifts of z = 0 , . , .
55 and 1 .
01. We focus on parent halos identified by the ROCKSTAR algorithm and excludeany subhalos in the following analyses. The halo position is defined by the center-of-mass location of a subset of memberparticles in the inner halo density, while the velocity is computed by the average particle velocity within the innermost10% of the virial radius. We keep the halos with
M > . h − M (cid:12) as a very conservative choice to study the haloproperties (i.e. the smallest halos in the analysis consist of ∼ M [ h − M (cid:12) ] = 10 . − , − . , . − , − . and 10 . − . Table 2 summarizes the number of dark matter halos in each subgroup of interest. We use thesesubgroups to calibrate the model parameters as in Section 4.3.2. Mock galaxy catalogs
To test our model of the pairwise velocity distribution of dark matter halos, we produce a set of mock galaxy catalogs.For the simplest model, we consider a mass-limited sample with the halo mass above M th at different redshifts. For themass-limited sample, we consider two different mass thresholds of M th = 10 . and 10 . h − M (cid:12) , which are typicalhalo masses of massive early-type galaxies at z < (cid:104) N gal (cid:105) M , gives the mean number of galaxiesin host halos with mass M . As a representative example, we consider the spectroscopic sample of massive galaxies inthe SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS). There are two targets of galaxies in the BOSS, but wefocus on the sample referred to as CMASS. The CMASS sample is designed to be a roughly volume-limited sampleof massive, luminous galaxies (Masters et al. 2011) and has a large galaxy bias of b ∼
2, showing that most galaxiesreside in the dark matter halos of M ∼ h − M (cid:12) (White et al. 2011). The data are available at https://hpc.imit.chiba-u.jp/ ∼ nngc/. http://cosmo.nyu.edu/roman/2LPT/ http://lambda.gsfc.nasa.gov/toolbox/tbcambform.cfm Shirasaki et al.
For the HOD of the CMASS sample, we adopt the model in Reid et al. (2014) with the form of (cid:104) N gal (cid:105) M = (cid:104) N cen (cid:105) M + (cid:104) N sat (cid:105) M , (25) (cid:104) N cen (cid:105) M = 12 (cid:20) (cid:18) log M − log M min σ log M (cid:19)(cid:21) , (26) (cid:104) N sat (cid:105) M = (cid:104) N cen (cid:105) M (cid:18) M − M cut M (cid:19) α M H ( M − M cut ) , (27)where H ( x ) is the Heaviside step function, (cid:104) N cen (cid:105) M and (cid:104) N sat (cid:105) M represent the HODs for the central and the satellitegalaxies, respectively. We adopt the best-fit parameters in Reid et al. (2014): log M min = 13 . σ log M = 0 . M cut = 13 .
27, log M = 14 .
08, and α M = 0 .
76. Using the HOD in Eqs. (25)-(27), we populate the ν GC-Lhalos with hypothetical CMASS galaxies at z = 0 .
55 in the following manner.(i) We populate halos with central CMASS galaxies by randomly selecting halos according to the probability dis-tribution, (cid:104) N cen (cid:105) M (Eq. [26]). In this step, we assume that each central galaxy resides at the halo center.(ii) When halos have central galaxies, we then randomly populate the halos with satellite galaxies assuming a Poissondistribution with the mean of λ M = [( M − M cut ) /M ] α M H ( M − M cut ). We assume that the radial distributionof satellites on average follows that of dark matter in each host halo. We simply assume the analytical Navarro-Frenk-White (NFW) profile (Navarro et al. 1996), where we use the concentration-mass-redshift relation inDiemer & Kravtsov (2015), to compute the density profile for each host halo. We set the halo-centric radius ofeach satellite by drawing a random variable q which follows M NFW ( < q ) /M NFW ( < R ). Here M NFW ( < r )represents the enclosed mass predicted by the NFW profile as a function of radius r .(iii) For each satellite galaxy, we assign a virial random motion by using a Gaussian random variable with the zeromean and the variance of σ = (1 + z ) GM/ (2 R ). Note that the halo radius is defined in the comovingcoordinate in this paper.Note that in principle we could use the concentration of individual halos measured by ROCKSTAR to set the NFWdistribution of satellites. Instead we simply adopt the model of Diemer & Kravtsov (2015) to ignore a possible impactof scatter in the halo concentration. Our primary purpose is to validate if our model can be suitable to model thetwo-point correlation function in redshift space at a scale of ∼ h − Mpc where the dominant contribution to anyclustering observable is expected to come from pairs of central galaxies (e.g. Zheng & Guo 2016).3.3.
Clustering statistics
For a given catalog of mock galaxies in Section 3.2, we perform a two-point clustering analysis in redshift space totest if our model is useful for the most widely used statistics in redshift surveys. The two-point correlation functionof galaxies is formally defined by (cid:104) n g ( r ) n g ( r ) (cid:105) = ¯ n (1 + ξ gg ( r )) , (28)where r = r − r , n g represents the number density field of galaxies of interest, ¯ n g is the mean density, and ξ gg ( r ) isthe two-point correlation function. Since distances to individual galaxies are affected by the redshift-space distortion,the galaxy two-point correlation must be anisotropic as in Eq. (3) in practice. Eq. (3) also shows that the anisotropyin the observed galaxy clustering is set by the pairwise velocity PDF for a given cosmology. According to this fact, weshall validate our model of pairwise PDFs by studying the mapping of two-point correlation functions between realand redshift space.In this paper, we measure the two-point correlation functions, ξ S gg and ξ gg for a given mock catalog. For comparison,we then predict ξ S gg by using Eq. (3) with the true ξ gg in the simulations and our model of P g ( v z ). We derive ananalytic expression of P g ( v z ) for the HOD-based model in Appendix B. In the simulations, we adopt the distant-observer approximation and set the z -axis in the simulation to be the line-of-sight direction. We measure the twopoint correlation function by using the natural estimator of DD / RR − . In the periodic box without boundaries, we The Landy-Szalay estimator (Landy & Szalay 1993) is often adopted in the literature and has a different form from DD / RR −
1. However,the difference between two is only important for large scales of ∼
100 Mpc. Since we are working on smaller scales ( ∼
10 Mpc), it shouldnot be necessary for our purpose. airwise velocity between dark matter halos ξ gg ( r ), we employ the logarithmicbinning in the range of r = 0 . h − Mpc to 100 h − Mpc with the number of bins being 40. For the redshift-spacecorrelation, we measure ξ S gg ( s p , s π ) in the linearly-spaced bins over 0 < s p,π [ h − Mpc] <
50 with the number of binsin each direction being 50.In practice, it is more common to compress the information in the two-point correlation in redshift space by usingthe Legendre expansion: ξ (cid:96) ( s ) ≡ (cid:96) + 12 (cid:90) − d µ ξ S gg ( s p , s π ) L (cid:96) ( µ ) , (29)where s = ( s p + s π ) / , µ = s π /s , L (cid:96) ( µ ) is the Legendre polynomial of order (cid:96) . We measure the first three non-zeromoments ( (cid:96) = 0 , ,
4) for a given mock catalog. We evaluate ξ (cid:96) ( s ) by using the measurement of ξ S gg ( s p , s π ) with 20logarithmic bins in the range of 0 . < s [ h − Mpc] < . µ with the width of ∆ µ = 0 .
05. Forcomparison purposes, we estimate the variance of ξ (cid:96) ( s ) by dividing the data volume into 2 sub-volumes and measuring ξ (cid:96) ( s ) for galaxies in each sub-volume. We then compute the variance of ξ (cid:96) ( s ) at a given s asVar[ ξ (cid:96) ( s )] = V sub V full N − N (cid:88) i =1 (cid:2) ξ (cid:96) ( s ; i ) − ¯ ξ (cid:96) ( s ) (cid:3) , (30)where N = 8, V sub is the sub-volume, V full is the full data volume, ξ (cid:96) ( s ; i ) represents the clustering multipole for the i -th subsample, and ¯ ξ (cid:96) ( s ) is the average multipole over 8 sub-volumes. Note that V full = 8 V sub . Eq. (30) provides arough estimate of the sample variance of the measurements of ξ (cid:96) ( s ) for the survey volume of (1 . (cid:39) .
40 [ h − Gpc] . CALIBRATION OF THE MODEL PARAMETERSIn this section, we summarize how to calibrate the model parameters in Section 2 with the pairwise velocity statisticsin the simulations. In principle, we can determine the functions of ˜ ρ ( r, M , M , z ), ˜ ρ t ( r, M , M , r ) and ˜ ρ r ( r, M , M , r )by using the smoothed density distribution from N -body particles and the statistics of halo pairs. Unfortunately, theparticle data in the ν GC-L run are not saved because of the hard drive shortage. Hence, we assume the specific formsof three functions ˜ ρ ( r, M , M , z ), ˜ ρ t ( r, M , M , r ) and ˜ ρ r ( r, M , M , r ) as in Eqs (22) and (24), but we attempt tofind the parameters in the functions so that the model can reproduce the first three non-zero moments of the pairwisevelocity in the simulations.Given the joint PDF in Eq. (7), one can find the first three non-zero moments as (cid:104) v r (cid:105) ( r, M , M , z ) ≡ (cid:90) d v t d v r v r P ( v t , v r | r, M , M , z )= (cid:90) d δ µ r ( δ, r, z ) F ( δ | r, M , M , z ) , (31) σ t ( r, M , M , z ) ≡ (cid:90) d v t d v r v t P ( v t , v r | r, M , M , z )= (cid:90) d δ Σ t ( δ, r, M , M , z ) F ( δ | r, M , M , z ) , (32) σ r ( r, M , M , z ) ≡ (cid:90) d v t d v r v r P ( v t , v r | r, M , M , z ) − [ (cid:104) v r (cid:105) ( r, M , M , z )] = (cid:90) d δ Σ r ( δ, r, M , M , z ) F ( δ | r, M , M , z )+ (cid:90) d δ µ r ( δ, r, z ) F ( δ | r, M , M , z ) − [ (cid:104) v r (cid:105) ( r, M , M , z )] , (33)where ˜ ρ ( r, M , M , z ) sets the functional form of F , while ˜ ρ t,r ( r, M , M , z ) is involved in the functions of Σ t,r . Becauseour model assumes the functional form of µ r is known as Eq. (23), Eqs (31)-(33) may provide sufficient information todetermine the model parameters in ˜ ρ and ˜ ρ t,r . In this paper, we first find the parameters in ˜ ρ (i.e. A ρ , B ρ , and C ρ in Eq. [22]) by the least square fitting of the profile (cid:104) v r (cid:105) ( r, M , M , z ) for a given set of masses and redshift ( M , M ,and z ). After finding the best-fit values of A ρ , B ρ , and C ρ by fitting of (cid:104) v r (cid:105) ( r ), we then find the best-fit parameters inEq. (24) by comparing the profiles of σ t ( r ) and σ r ( r ) with the predictions as in Eqs (32) and (33).0 Shirasaki et al. r [ h − Mpc]100150200250300350 − h v r i [ k m / s ] A ρ Fiducial 10 r [ h − Mpc] B ρ × . r [ h − Mpc] C ρ × . Figure 1.
Dependence of the mean radial velocity profile on the model parameters in Section 2. In each panel, the solid lineshows the profile when we adopt the parameters proposed in Tinker (2007), while the dashed (dotted) line stand for the caseswhen varying a given parameter by a factor of 1.2 (0.8). The left, middle, right panels show the dependence on A ρ , B ρ , and C ρ ,respectively. In this figure, we consider the halo pairs of M = M = 10 h − M (cid:12) at z = 0. Figure 1 shows the model prediction of the mean radial velocity profile as a function of model parameters A ρ , B ρ ,and C ρ in Eq. (22). In the figure, we consider a pair of halos with their masses of M = M = 10 h − M (cid:12) . Thefigure represents a good flexibility of our model for fitting of (cid:104) v r (cid:105) . For the velocity dispersions, we fix four parametersin the fitting process to avoid a complex degeneracy among parameters. To be specific, we fix C (2) t = 0 . q t = − . p r = − .
0, and q r = − . σ t and σ r ( r ) in the simulation.In the fitting processes, we measure the three moments of the pairwise velocity for the samples in Table 2. For themeasurements, we employ a linear-space binning in the range of 0 < r [ h − Mpc] <
40 with 200 bins. For a given halosample, we then find the best-fit values of A ρ , B ρ and C ρ in a given M , M , and z bin by minimizing the following χ statistic: χ ( p mean | M , M , z ) = (cid:88) i [ (cid:104) v r, sim (cid:105) ( r i ) − (cid:104) v r, mod (cid:105) ( r i | , p mean )] σ r, sim ( r i ) /N pairs ( r i ) , (34)where (cid:104) v r, sim (cid:105) ( r i ) is the mean radial velocity profile at the i -th radius in the simulation, (cid:104) v r, mod (cid:105) ( r i ) is the counterpartof our model prediction, p mean = ( A ρ , B ρ , C ρ ), σ r, sim ( r i ) is the dispersion of v r at the i -th radius, and N pair ( r i ) is thenumber of pairs in the i -th radius. Once the best-fit p mean is found, we then minimize other χ quantities to find thebest-fit parameters in Eq. (24): χ ( p α | M , M , z ) = (cid:88) i (cid:104) σ α, sim ( r i ) − σ α, mod ( r i | , p α ) (cid:105) σ α, sim ( r i ) /N pairs ( r i ) , (35)where α = t or r , σ α, sim ( r i ) is the velocity dispersion profile at the i -th bin radius, σ α, mod ( r i ) is the our model prediction, p t = ( C (0) t , C (1) t , p t ) and p r = ( C (0) r , C (1) r , C (2) r ).Given the sets of p mean , p t , p r as a function of M , M , and z , we then find an appropriate form to smoothlyinterpolate the data points after trial and error. For an example, we assume that the form of B ρ ( M , M , z ) is given by B ( z ) [( M + M ) / h − M (cid:12) ] B ( z ) . We then find the best-fit B and B for a given z by a least-square fitting withthe measured B ρ ( M , M , z ). The redshift dependence of B and B is then derived by a quadratic function fit. Thedetails of our functional forms for other parameters are provided in Appendix A . It is obvious that our calibration We also make our pipeline for the calibration process publicly available at https://github.com/shirasakim/Fitting velocity moments T07. airwise velocity between dark matter halos − − − h v r i [ k m / s ] Our model10 . − [ h − M (cid:12) ]10 − . [ h − M (cid:12) ]10 . − [ h − M (cid:12) ] 0100200300400 σ t + c o n s t . [ k m / s ] M = M at z = 0 . σ r + c o n s t . [ k m / s ] r [ h − Mpc]0 . . . S i m / M o d e l r [ h − Mpc]0 . . . r [ h − Mpc]0 . . . Figure 2.
The mean and dispersions of pairwise velocities of dark matter halos at z = 0 .
55. We here focus on the halosamples with M = M = M = 10 . − , − . , and 10 . − h − M (cid:12) . In each panel, the blue circle, orange sqaure, andgreen plus symbols represent the simulation results for M = 10 . − , − . , and 10 . − h − M (cid:12) , respectively. The modelpredictions are shown in the solid lines. The error bar in each panel shows the Gaussian error at a given radius. The upperthree panels present the profiles of mean radial velocity, the dispersions of the tangential and radial components, while the lowerpanels show the ratio between the simulation results and our model prediction. Note that the Gaussian error in each panel istoo small to plot for most cases. For visualization purpose, we shift the profiles of σ t,r by −
100 and +100 km / s for the sampleof M = 10 . − and 10 . − h − M (cid:12) , respectively. process can be affected by details of the interpolation of p mean , p t , and p r (e.g. a choice of the functional form). Thispoint is discussed in Section 6.3. RESULTSHere we present the comparison of the pairwise velocity statistics from our model with the simulation results. Wepay a special attention to the results at z = 0 .
55, because it is relevant to the CMASS sample in the SDSS-III BOSS.We then discuss the information content in redshift-space clustering in terms of the pairwise velocity statistics.5.1.
Velocity moments for galaxy- and group-sized halos
We first show the results of the first three non-zero moments of the pairwise velocity for the halo samples with theirmasses of 10 . − , − . , and 10 . − h − M (cid:12) . Figure 2 summarizes the comparisons of (cid:104) v r (cid:105) , σ t , and σ r fromour model with the simulation results. In the range of 5 < ∼ r [ h − Mpc] <
40, our model can reproduce the mean anddispersion of the pairwise velocity of the simulated halos with 10 . < M [ h − M (cid:12) ] < . within a 5%-level precision,while still providing a reasonable fit to the results for group-sized halos with 10 . < M [ h − M (cid:12) ] < . Accordingto the Gaussian error estimate, our measurement of velocity moments in simulations is precise with a level of < M < h − M (cid:12) and z <
1. The number of halo pairs including halos of
M > h − M (cid:12) at z = 1 .
01 becomessmall, but our measurements reach a 15%-level precision even at halo pairs with their masses of < . h − M (cid:12) at z = 1 . z = 0 . r < h − Mpc]. For other redshifts and halo masses, we summarize thecomparisons in Appendix C. We confirm that our model can reproduce the velocity-moment profiles with a 5%-levelprecision for 10 . < M [ h − M (cid:12) ] < . at 0 . < z <
1, and the model precision reaches a 20% level at the veryworst in other ranges of halo masses and redshift.2
Shirasaki et al.
Mass-limited samples
We then move onto the comparisons of various velocity statistics for the mass-limited halo samples. Because ourcalibration process includes the interpolation of model parameters as a function of halo mass and redshift, it isimportant to check if our model still works for samples with a wider range of halo masses.5.2.1.
Pairwise-velocity distribution and its moments
Figure 3 shows the comparisons of the velocity PDFs in the simulation with our model prediction. In each panel, thegray and black solid lines represent the model predictions for v r and v t , respectively. For a comparison, we show thepredictions by the model in Tinker (2007) by the gray and black dashed lines. Although our model has been calibratedby the measurements of the three velocity moments as in Eqs. (31)-(33), the non-Gausssian tails in the PDFs can beexplained by our model in a reasonable way. This indicates that a large part of the non-Gaussianity in the velocityPDFs can be related with the non-Gaussianity in the cosmic mass density (see Tinker 2007, for further discussion).Compared to the previous work, our model can provide a better fit to the velocity dispersion of v t as well as thelong tails in PDFs for a wide range of r . We find that the mean and velocity dispersion can be explained by our modelwithin a 10% level for the mass-limited sample with M ≥ . h − M (cid:12) . This is clearly shown in Figure 4 and valid forthe redshifts of z = 0 . M ≥ . h − M (cid:12) , we find that our model is ingood agreement with the simulation results and the precision reaches a 5% level in the range of 5 < r [ h − Mpc] < Two-point clustering analyses in redshift space
We next examine a more practical analysis of the two-point correlation function in redshift space, denoted by ξ S gg ( s p , s π ), for the mass-limited samples. Note that we suppose the real-space correlation function is known in thispaper, while we study the mapping of the correlation function between real and redshift space by our model ofthe velocity PDFs (see Eq. [3]). The details about modeling of the velocity PDF with a HOD are summarized inAppendix B.Figure 5 shows the comparisons of ξ S gg between the simulation results and the model predictions. For the modelprediction of ξ S gg , we use the real-space correlation function measured in the simulations and interpolate the data pointsover separation lengths r . We find that our model can provide a more reasonable fit to ξ S gg in the simulations thanthe models by Tinker (2007). In particular, our model improves the mapping at small s π , because our model shows abetter fit to the velocity dispersion of v t in the simulations and v t is relevant to the line-of-sight velocity at small s π .For a more quantitative view, we show comparisons of the clustering multipoles defined in Eq. (29). Figure 6summarizes comparisons for the mass-limited sample with M ≥ . h − M (cid:12) . The figure clearly shows that ourmodel can provide an accurate mapping of the two-point correlation function between real and redshift space atintermediate scales of ∼ h − Mpc. For the lowest-order moment, our model can provide an excellent fit to thesimulation results within a 5% level over 5 < s [ h − Mpc] <
40. Even for the higher-order moments, we find that ourmodel can explain ξ ( s ) and ξ ( s ) in the range of 10 − h − Mpc with a 10%- and 50%-level precision, respectively.For the mass-limited sample with M ≥ . h − M (cid:12) , the agreement is found to be worse compared to the sampleswith M ≥ . h − M (cid:12) . Nevertheless, the lowest moment ξ can still be reproduced by our model within a 5% levelprecision over 3 < s [ h − Mpc] <
30 even for the sample with M ≥ . h − M (cid:12) .5.3. Realistic galaxy samples
Here we present the results for a realistic galaxy mock sample with the HOD model in Section 3.2. Figure 7summarizes the comparisons of the velocity-moment profiles in the simulation with our model predictions. We findthat our model can provide a few-percent-level prediction of the mean radial velocity as well as the velocity dispersionsfor CMASS-like galaxies at 5 − h − Mpc.For the clustering multipoles, figure 8 shows the comparisons between the simulation results and our model pre-dictions. The red solid lines in the figure represent our model predictions, while the dashed line shows the so-calledtwo-halo terms in a halo-model approach (see Appendix B for details). In our halo model, the redshift-space clusteringmultipoles can be decomposed into two parts: ξ (cid:96) ( s ) = ξ (cid:96), ( s ) + ξ (cid:96), ( s ) , (36)where ξ (cid:96), represents the two-point correlation in single dark matter halos, ξ (cid:96), is the contribution from the clusteringbetween two neighboring halos. The one-halo term ξ (cid:96), can be further divided into two contributions from the central-satellite and satellite-satellite pairs. The two-halo term ξ (cid:96), is mostly determined by the clustering and streaming airwise velocity between dark matter halos − − − − f ( v t / r ) × r = 3 . h − Mpc v t (Sim) v r (Sim) r = 6 . h − Mpc M ≥ . [ h − M (cid:12) ] at z =0.55 v t (Tinker 2007) v r (Tinker 2007) r = 9 . h − Mpc v t (Our model) v r (Our model) − − − − f ( v t / r ) × r = 12 . h − Mpc r = 15 . h − Mpc r = 18 . h − Mpc − v t/r [km / s]10 − − − − f ( v t / r ) × r = 21 . h − Mpc − v t/r [km / s] r = 24 . h − Mpc − v t/r [km / s] r = 27 . h − Mpc
Figure 3.
The probability distribution functions (PDFs) of the pairwise velocity of dark matter halos with their massesgreater than 10 . h − M (cid:12) at the redshift of 0 .
55. The nine panels show the PDFs at different separation length, r (labeledat the top right in each panel). In each panel, the blue circle and orange square symbols show the PDFs of v t and v r in thesimulation, respectively. Our model predicitons are shown in the black and gray solid lines in each panel, while the dashed linesrepresent the models in Tinker (2007). Our model improves the precision for the velocity dispersion of v t as well as the meanvelocity of v r compared to the model in Tinker (2007). Note that we do not introduce any parameters to explain the skewnessand kurtosis in our model, and any non-Gaussian features in the PDFs come from non-Gaussianity in the cosmic mass density(see Section 2 for our model). Shirasaki et al. − − − − − h v r i [ k m / s ] M ≥ . [ h − M (cid:12) ] (Sim) σ t [ k m / s ] z = 0 . M ≥ . [ h − M (cid:12) ] (Tinker 2007) σ r [ k m / s ] M ≥ . [ h − M (cid:12) ] (Our model) r [ h − Mpc]0 . . . S i m / O u r m o d e l r [ h − Mpc]0 . . . r [ h − Mpc]0 . . . Figure 4.
Similar to Figure 2, but for a mass-limited sample of halos. Here we plot the mean and dispersion of pairwise-velocity of dark matter halos with their masses greater than 10 . h − M (cid:12) at the redshift of 0 .
55. In the upper three panels,the blue circles show the simulation results, while the solid and dashed lines represent the predictions by our model and Tinker(2007), respectively. In the lower panels, we show the ratio between the simulation result and our model. For a reference, thegray regions show ±
10% levels, while the yellow one stands for ±
5% levels.
10 20 s p [ h − Mpc]5101520 s π [ h − M p c ] M ≥ . [ h − M (cid:12) ] at z =0.55 SimTinker (2007)Our model
10 20 s p [ h − Mpc]5101520 s π [ h − M p c ] M ≥ . [ h − M (cid:12) ] at z =0.55 SimTinker (2007)Our model
Figure 5.
Two-dimensional redshift-space correlation functions of mass-limited halo samples at z = 0 .
55. The left panelshows the result for M ≥ . h − M (cid:12) , while the right stands for M ≥ . h − M (cid:12) . In each panel, the gray lines show thecontours of the redshift-space correlation of ξ S gg ( s p , s π ), and the orange dashed lines are the model predictions by our model.For a comparison, the blue sold lines shows the model by Tinker (2007). In each panel, the contours are separated by factorsof 1.9 for clarity. The outermost represents ξ S gg ( s p , s π ) = 1 . − . and 1 . − . in the left and right panels, respectively. motion of the central-central pairs, but it is also affected by the velocity dispersion of satellites in single halos. Ourmodel predictions are in good agreement with the simulation results at the scales of s > ∼ h − Mpc, i.e. the regimewhere the two-halo contributions would play a central role. Note that the inaccurate small-scale two-halo term canaffect the prediction of ξ ( s ) at s ∼ − h − Mpc. We find that a simple modification in the two halo term of ξ can airwise velocity between dark matter halos s ξ [ ( h − M p c ) ] M ≥ . [ h − M (cid:12) ] (Tinker 2007) M ≥ . [ h − M (cid:12) ] (Our model) M ≥ . [ h − M (cid:12) ] (Sim) − − s ξ [ ( h − M p c ) ] z = 0 . − s ξ [ ( h − M p c ) ] s [ h − Mpc]0 . . . S i m / O u r m o d e l s [ h − Mpc]0 . . . s [ h − Mpc]0 . . . Figure 6.
Redshift-space clustering multipoles of the mass-limited halo sample with M ≥ . h − M (cid:12) at z = 0 .
55. Theupper panels show the multipole moments of ξ , ξ and ξ from left to right. In each upper panel, the blue points with errorbars show the simulation results, while the solid and dashed lines represent the predictions by our model and Tinker (2007),respectively. In the lower panels, we show the ratio of the simulation results and our model predictions. The yellow and grayfilled regions show ± ± − − − − − h v r i [ k m / s ] Our modelCMASS-like mock 200250300350400450 σ t [ k m / s ] z = 0 . σ r [ k m / s ] r [ h − Mpc]0 . . . S i m / O u r m o d e l r [ h − Mpc]0 . . . r [ h − Mpc]0 . . . Figure 7.
Similar to Figure 4, but for a mock sample of CMASS galaxies at z = 0 .
55. In each panel, the blue points show thesimulation results, while the solid lines represent our model predictions. Note that the typical halo mass of CMASS galaxies isset to ∼ h − M (cid:12) , but we include the satellite galaxies for M > . h − M (cid:12) . provide a better fit to the simulation: ξ , ( s ) → exp (cid:20) − (cid:16) s cut s (cid:17) (cid:21) ξ , ( s ) , (37)where s cut is a free parameter and we find s cut ∼ . h − Mpc is appropriate for our mock catalog. The two-haloterm ξ , at s < h − Mpc would be affected by the non-Gaussianity in the pairwise velocity PDFs (e.g. Cuesta-6
Shirasaki et al. s ξ [ ( h − M p c ) ] − − − s ξ [ ( h − M p c ) ] z = 0 . s ξ [ ( h − M p c ) ] s [ h − Mpc]0 . . . S i m / O u r m o d e l s [ h − Mpc]0 . . . s [ h − Mpc]0 . . . Figure 8.
Similar to Figure 6, but for a mock sample of CMASS galaxies at z = 0 .
55. In each panel, the blue points witherror bars are the simulation results, while the red solid line shows the model predictions based on the HOD and our model ofthe pairwise velocity PDFs. For a comparison, the dashed lines show the two-halo contribution to the multipole moments.
Lazaro et al. 2020). Our model has been calibrated to explain the mean and variance in the pairwise velocity PDF at5 < r [ h − Mpc] <
40. After the calibration, we found that it fails to provide a fit to the PDF at r < h − Mpc (e.g.see the top left panel in Figure 3). Hence, we expect that our model can not work for the precise prediction of ξ , at s < h − Mpc. 5.4.
Information contents of redshift-space clustering multipoles
For an application of our model, we discuss the information content in redshift-space clustering analyses of galaxies.According to Eq. (3), the observed two-point correlation function in redshift space should contain the informationabout the pairwise velocity statistics of galaxies. At intermediate scales of ∼
10 Mpc, a class of modified gravitytheory predicts that the mean and dispersion of the pairwise velocity for massive-galaxy-sized halos can differ fromthe prediction from General Relativity by 10 −
20% (e.g. Hellwing et al. 2014; Zu et al. 2014). For the galaxy-haloconnection, numerical simulations have shown that the pairwise velocity statistics for realistic galaxies can depend notonly on their host halo masses but also the inner mass density profiles of their host halos and ages (e.g. Hearin 2015;Padilla et al. 2019). This is known as the assembly bias effect.As a simple example, we study the modified gravity effect and/or the assembly bias effect on the halo pairwisevelocity by introducing two free parameters: v obs z = b v (cid:104) v z (cid:105) + b v ( v z − (cid:104) v z (cid:105) ) , (38)where v obs z represents the line-of-sight pairwise velocity affected by the modified gravity and/or the assembly biaseffect, while v z is a baseline prediction in the ΛCDM cosmology. We here assume that statistical properties of v z canbe characterized by halo masses, redshifts, and separation lengths. Note that b v changes the mean pairwise velocityfor a given galaxy sample, while b v affects the variance in the pairwise velocity. The modified gravity and/or theassembly bias can deviate b v and b v from unity. Therefore, it would be interesting to consider the dependence of theclustering multipoles of ξ , on the velocity-bias parameters b v and b v . We here emphasize that our velocity biases inEq. (38) have a different meaning from the common definitions in the literature. Previous studies have mainly focusedon the velocity biases with respect to the core or the kinematics of dark matter inside the single dark matter halo (e.g.Reid et al. 2014; Guo et al. 2015). In contrast, our parameterization of the velocity bias enables us to study the bias in airwise velocity between dark matter halos − s ξ ‘ + c o n s t . [ ( h − M p c ) ] s ξ + 50 s ξ log M min Fiducial − . . b v Fiducial − . . b v Fiducial − . . . . . R a t i o ( ξ ) s [ h − Mpc]0 . . . R a t i o ( ξ ) s [ h − Mpc] 10 s [ h − Mpc]
Figure 9.
Dependence of redshift-space clustering multipoles on a HOD parameter and velocity biases. In the three top panel,we show the monopole ξ and quadropole ξ for the CMASS-like mock galaxies at z = 0 .
55 when varying the HOD parameterlog M min and the velocity-bias parameters b v and b v from left to right. The definitions of b v and b v are found in Section 5.4.In each panel, the black solid lines show the results for our fiducial set of the parameters, while the blue dashed and red dottedlines represent the responses of ξ , when we vary the parameters. In the three middle panels, we summarize the ratio of ξ with respect to the fiducial results. We also show the ratio of ξ in the three bottom panels. For references, the gray and yellowfilled regions in the middle and bottom panels show ±
10- and ± the streaming motion between two neighboring halos . Using Eq. (38) and the formulas in Appendix B, we computethe expected signal of ξ , for the HOD model in Section 3.2 as a function of b v and b v . To check for degeneracyamong the HOD parameters, we also vary a parameter of log M min which determines the typical halo mass of galaxysample of interest. We generated two additional mock galaxy catalogs using the ν GC-L simulation by changing theparameter of log M min by ± .
05. In the following, we use these simulation results when studying the effect of log M min .When varying the biases of b v and b v , we use the analytic model of the pairwise velocity PDF as in Appendix B andpredict the multipoles based on Eq. (3). We also adopt the correction in Eq. (37) for our model of the two-halo termin ξ .Figure 9 summarizes the changes in ξ , for the CMASS-like galaxy sample at z = 0 .
55 caused by differences inlog M min , b v and b v . In the figure, we set the HOD parameters as in Section 3.2 and b v = b v = 1 for the fiducialcase. The figure indicates that the effect of b v and b v on the redshift-space clustering can not be compensated for by Within our framework, non-trivial galaxy-halo connection may induce biases in the streaming motion between two galaxies. An exampleis the environmental dependence of HODs (e.g. Hadzhiyska et al. 2020). If the HOD depends not only on the halo mass but also theenvironmental density δ , the pairwise velocity statistics can differ from our predictions. We also expect that a modification of gravity canchange the relation of the mean infall velocity and density perturbations (see, e.g. Li & Efstathiou 2012, for the spherical collapse modelin a modified gravity theory), leading to b v (cid:54) = 1. Shirasaki et al. − − − − − h v r i [ k m / s ] Model (Planck)Sim (Planck) σ t [ k m / s ] CMASS-like HOD
Model (WMAP5)Sim (WMAP5) σ r [ k m / s ] r [ h − Mpc]0 . . . S i m / O u r m o d e l r [ h − Mpc]0 . . . r [ h − Mpc]0 . . . Planck WMAP5
Figure 10.
Similar to Figure 7, but we include the comparison with the results from the MultiDark simulation at z = 0 . m0 = 0 .
31 and σ = 0 . m0 = 0 .
27 and σ = 0 .
82. In the upper panels, the blue circle and pink square symbolsrepresent the simulation results in the Planck and WMAP5 cosmology, respectively. The solid line shows our model predictionfor the Planck cosmology, while the red dashed line is the model prediction for the WMAP5 cosmology. simple changes in the typical host halo mass. When it comes to other HOD parameters, we find that σ log M and M show a strong degeneracy with log M min , while M cut and α M can change the one-halo term while changing two-haloterm minimally. We also note that the real-space correlation function strongly depends on the HOD parameters, butis independent of the biases of b v and b v . The real-space correlation function can be reproduced within the HODframework for a given cosmological model and the HOD parameters have been tightly constrained with the combinedanalysis of galaxy-galaxy lensing and projected correlation functions (e.g. More et al. 2015). Therefore, we expect thata joint analysis of ξ , with galaxy-galaxy lensing and projected correlation functions provides an important test ofthe common HOD framework with no assembly biases at least. For more details (e.g. expected constraints of b v and b v for a given galaxy sample), we require a precise estimate of the covariance of ξ , and leave it for future studies. LIMITATIONSWe summarize the major limitations in our model of pairwise velocity PDFs of dark matter halos. All of the followingissues will be addressed in forthcoming studies.6.1.
Cosmological dependence
Our model of pairwise velocity PDFs is calibrated against N -body simulations in the ΛCDM cosmology consistentwith Planck. In terms of studies of large-scale structure, Ω m0 and σ are the primary parameters and the simulationsin this paper adopt Ω m0 = 0 .
31 and σ = 0 .
83. Therefore, our functions in Section 2 and Appendix A may be subjectto an overfitting to the specific cosmological model. To examine the dependence of our model on cosmological models,we use another halo catalog from N -body simulations with a different ΛCDM model. For this purpose, we use thefirst MultiDark simulation performed in Prada et al. (2012). The MultiDark simulation consists of 2048 particles in avolume of 1 [ h − Gpc] and assumes the cosmological parameters of Ω m0 = 0 .
27, Ω b0 = 0 . Λ = 1 − Ω m0 = 0 . h = 0 . n s = 0 .
95, and σ = 0 .
82. These are consistent with the five-year observation of the cosmic microwavebackground obtained by the WMAP satellite (Komatsu et al. 2009) and we refer to them as the WMAP5 cosmology.We use the ROCKSTAR halo catalog at z = 0 .
534 from the MultiDark simulation and then produce a CMASS-like The halo catalogs at different redshifts are publicly available at https://slac.stanford.edu/ ∼ behroozi/MultiDark Hlists Rockstar/. airwise velocity between dark matter halos − h − Mpc. It is worth noting that the cosmological dependence of the velocity dispersion is smallin the simulations, but our model predicts a few percent level difference. For the mean radial velocity profile, wefind a 10%-level difference between the simulation and our model in the WMAP5 cosmology at r < h − Mpc, whileour model provides a better fit to the simulations at larger scales of r > ∼ h − Mpc. For comparison, our model canpredict the mean radial velocity profile at r = 5 − h − Mpc within a 5%-level precision in the Planck cosmology.In summary, our model can not predict the simulation results for the WMAP5 cosmology with the same level as inthe Planck cosmology. The 10%-level difference in Ω m0 can cause systematic uncertainties in our model predictionswith a level of 5-10%. Note that the velocity dispersions are found to be less sensitive to the change in Ω m0 in thesimulations. More extensive studies are required to investigate the cosmological dependence of the pairwise velocitystatistics. 6.2. Calibrations with N -body particle data Our model assumes that the pairwise velocity PDFs can be expressed as a Gaussian at a given environmental density δ . Tinker (2007) already showed that the approximation looks valid by using the N -body simulations, while the δ dependence of the Gaussian parameters (mean and variance) may be different from our assumptions in Eqs. (18) and(23). We also assume that the conditional PDF of cosmic mass density finding a halo pair is given by the form ofEq. (8), but another functional form would provide a better fit to the simulation results at small scales. The calibrationwith the information of N -body particles is important to validate the underlying assumptions in our model.After the calibration, we find that our model can not provide a reasonable fit to the pairwise velocity PDF ofmass-limited halos at z = 0. Figure 11 summarizes the pairwise velocity PDFs for the mass-limited sample with M ≥ . h − M (cid:12) at z = 0. The figure shows sizable differences of radial velocity PDFs between our model and thesimulation results. Note that the standard deviation σ r,t can be explained by our model within a 5%-level precision forthis mass-limited sample, but the mean radial velocity profiles in the simulation are larger than our model predictionsby (cid:39) N -body particles allows us to find more appropriate functional forms. Also, our modelcan not explain the velocity-moment profiles at r < ∼ h − Mpc for most cases. This also implies that Eqs. (18) and(23) may need some corrections for the velocity statistics at r < ∼ h − Mpc. Note that the log-normal approximationfor the cosmic mass density PDF can be less accurate at the scale of r < ∼ h − Mpc (e.g. Shin et al. 2017).6.3.
Interpolation errors and more precise modeling
As in Section 4, our calibration is based on the least square fitting of velocity-moment profiles and the interpolationof the best-fit parameters over halo masses and redshifts. Our interpolation scheme provides the best performance fordark matter halos with 10 . < M [ h − M (cid:12) ] < . at 0 . < z <
1, but it gives less precise predictions for otherranges of M and z . Figure 12 summarizes an example of the interpolation error in our calibration process. In thisfigure, we show the velocity-moment profiles for halos with M = 10 . − h − M (cid:12) . After the fitting process, we findthe best-fit expression of each profile as shown in the orange lines of Figure 12. Since the final model involves withinterpolation of model parameters over M and z , sizable residuals can be found in the comparisons of velocity-momentprofiles if we use an inaccurate interpolation method. Figure 12 also highlights that the best-fit expression reaches afew-percent-level precision for a given bin of masses and redshift. This indicates that a more sophisticated interpolationbeyond the use of an analytic function will further improve the precision of our model prediction.A promising approach for the interpolation of our model parameters is the Gaussian Process Regression. TheGaussian Process Regression allows to interpolate a large-dimensional dataset in a non-parametric way and it isbecoming a standard approach to develop accurate models for various statistics of large-scale structures (e.g. Habibet al. 2007; Lawrence et al. 2010; Kwan et al. 2013, 2015; McClintock et al. 2019; Nishimichi et al. 2019). For theGaussian Process Regression, one usually needs to reduce the effective numbers of data points in some way such as0 Shirasaki et al. − − − − f ( v t / r ) × r = 3 . h − Mpc v t (Sim) v r (Sim) r = 6 . h − Mpc M ≥ . [ h − M (cid:12) ] at z =0.00 v t (Tinker 2007) v r (Tinker 2007) r = 9 . h − Mpc v t (Our model) v r (Our model) − − − − f ( v t / r ) × r = 12 . h − Mpc r = 15 . h − Mpc r = 18 . h − Mpc − v t/r [km / s]10 − − − − f ( v t / r ) × r = 21 . h − Mpc − v t/r [km / s] r = 24 . h − Mpc − v t/r [km / s] r = 27 . h − Mpc
Figure 11.
Similar to Figure 3, but we here show the probability distribution functions (PDFs) of the pairwise velocity ofdark matter halos with their masses greater than 10 . h − M (cid:12) at the redshift of 0. the Principle Component Analysis. In our approach, we can reduce the number of model parameters in a physically-motivated way. Our analyses show that only three functions, ˜ ρ ( r, M , M , z ), ˜ ρ t ( r, M , M , z ) and ˜ ρ r ( r, M , M , z )(see Section 2 for details) will be sufficient to fit the pairwise velocity PDFs for the Planck cosmology.It would be worth mentioning that this is a huge reduction of the number of dimensions in the model compared toother models of pairwise velocity PDFs in the literature. Zu & Weinberg (2013) introduced a two-dimensional skewed-tdistribution with seven functions to explain the pairwise velocity PDFs of galaxies around clusters. The seven functions airwise velocity between dark matter halos − − − − − h v r i [ k m / s ] ModelBest fitSim 200250300350 σ t [ k m / s ] . ≤ M , M [ h − M (cid:12) ] < at z = 0 .
55 250300350400 σ r [ k m / s ] r [ h − Mpc]0 . . . R a t i o r [ h − Mpc]0 . . . r [ h − Mpc]0 . . . Figure 12.
Interpolation error in our model of the pairwise velocity PDFs. In the upper panels, the blue points show thevelocity-moment profiles of (cid:104) v r (cid:105) , σ t , and σ r for the halos with M = M = 10 . − h − M (cid:12) at z = 0 .
55, while the solid lines areour model predictions. The orange line shows the best-fit model inferred by the least-square fitting of velocity moment profilesin the calibration process (see Section 4 for details). In the bottom, we show the ratio of profiles between simulation resultsand our model with the blue points, while the orange line shows the ratio between the best-fit and our model which includesthe interpolation over halo masses. in Zu & Weinberg (2013) depend on r, M , M and z in principle. Bianchi et al. (2016) developed a model of the pairwisevelocity PDFs which is valid for both dark matter particles and halos. The model requires the knowledge of the firstthree moments of the line-of-sight pairwise velocity distribution plus two well-defined dimensionless parameters, andeach is a function of r, M , M and z . Kuruvilla & Porciani (2018) found that a mixture of Gaussian PDFs can providean excellent fit to the pairwise velocity PDFs for the line-of-sight component in the simulations. This model requiresfive functions to set full properties of the velocity PDFs. These five functions are dependent on r p , r π , M , M and z for dark matter halos. Recently, Cuesta-Lazaro et al. (2020) proposed that a one-dimensional skewed-t PDF canprovide a sufficient fit to the PDFs of the line-of-sight pairwise velocity for dark matter halos with M ≥ h − M (cid:12) at z = 0. A skewed-t PDF has four free parameters and each will depend on r p , r π , M , M and z in general. Mostprevious studies have not studied the dependence of their PDF model on halo masses, redshifts, and the separationlengths. Future studies should focus on efficient calibrations and emulations of the mass-redshift-scale dependence ofpairwise velocity PDFs for dark matter halos. DISCUSSION AND CONCLUSIONIn this paper, we developed a semi-analytic model of the pairwise velocity distributions of dark matter halos. Themodel is motivated by the findings and framework in Tinker (2007) and we re-calibrated the model parameters in therelation between the pairwise velocity and an environmental density around halo pairs using high-resolution N -bodysimulation covering a volume of ∼ .Our model has three functions related to the halo formation and the dependence of velocity dispersions on thecosmic mass density. By combining the log-normal PDF of cosmic mass density, our model can realize a significantnon-Gaussianity in the pairwise velocity PDF with three parameters alone, while previous non-Gaussian PDF modelsrequire more parameters. We calibrated these three as a function of halo masses ( M and M ), redshifts z andthe separation lengths r using halo catalogs for 10 . < [ h − M (cid:12) ] < and 0 < z <
1. We found that ourmodel can reproduce the first three non-zero velocity moments at 5 < r [ h − Mpc] <
40 for the halo masses of10 . < ∼ M [ h − M (cid:12) ] < ∼ . at 0 . < z < Shirasaki et al. M ≥ . h − M (cid:12) at z = 0 .
55, we confirmed that our model can explain the redshift-space clustering monopole andquadropole in the range of 5 −
40 and 10 − h − Mpc within a 5%-level precision. This is valid even for a realisticSDSS-III BOSS CMASS galaxy sample based on the framework of a halo occupation distribution (HOD), if we havean accurate model of the real-space correlation function of galaxies. We then studied the dependence of the clusteringmultipoles on the velocity biases in the galaxy straming motion by using our model. We found that a 20%-level biasin the mean and dispersion of the pairwise velocity of galaxies can induce a characteristic scale dependence of theobservables at ∼
10 Mpc. It would be difficult to reproduce these features by varying the typical halo mass of galaxiesalone, but more investigations are needed to make a robust conclusion.Although our model of the pairwise velocity PDFs will play an important role in cosmological analyses in redshiftsurveys of massive galaxies, we require further improvements of the model before applying it to real data sets. In fact,the statistical uncertainties of the redshift-space clustering monopole and quadropole for the massive galaxies in BOSSalready reach a level of a few percent at 1 − h − Mpc (e.g. Reid et al. 2014) and our model precision is comparableto them at best. To improve the model precision, we may require a more sophisticated approach to calibrate modelparameters such as Gaussian Process Regression, or some modifications in the functional forms in our model. Analysesinvolved with N -body particle data would be a key to improve our model, because the relationship between the cosmicmass density and the halo velocity is the essential part in our model. In addition, our model assumes the specificcosmological model in a ΛCDM scenario. We require further investigations to study the cosmological dependence ofour model as well as extend our framework to include modified gravity theories. Upcoming redshift surveys aim atmeasuring the redshift-space clustering of galaxies with lower masses and higher redshifts than the mass- and redshiftranges explored in this study. It is thus important to extend our approach so as to be applicable for a wider range ofhalo masses and redshifts.The model presented in this paper is an important first step toward statistical inference of the kinematics of galaxiesfrom their clustering information in redshift surveys as well as interpretation of the small-scale measurements ofthe kinematic Sunyaev-Zel’dovich effect. Precise analyses with current and upcoming redshift surveys enable us tostudy the motion of several tracers of large-scale structures. The kinematic information of the tracers can providean independent and important test of the standard cosmological model and allow us to examine possible deviationsfrom General Relativity, if we have an accurate model of the pairwise velocity PDFs of dark matter halos. Our futurework with the model of the pairwise velocity include a joint analysis of galaxy-galaxy lensing and the redshift-spaceclustering to infer the streaming motion of dark matter halos and investigation of the small-scale information in thekinematic Sunyaev-Zel’dovich effect on massive galaxies at various redshifts.ACKNOWLEDGMENTSWe thank the ν GC collaboration for making their simulation data publicly available. This work is in part supportedby MEXT KAKENHI Grant Number (18H04358, 19K14767). MS is supported by JSPS overseas Research Fellowshipsduring his stay at the Jet Propulsion Laboratory (JPL). Numerical computations were in part carried out on CrayXC50 at Center for Computational Astrophysics, National Astronomical Observatory of Japan. EH, KM, and JR weresupported by JPL, which is run by Caltech under a contract with the National Aeronautics and Space Administration(80NM0018D0004). APPENDIX A. LIST OF MODEL PARAMETERSIn this appendix, we provide the fitting functions in our model of the pairwise velocity distribution. The model issummarized in Section 2 and we introduce 58 parameters to explain the dependence of our model on halo masses,redshifts, separation lengths between halos.For Eq. (22), we find the following forms provide a reasonable fit to the simulation results: A ρ ( M , M , z ) = a ( z )[ D ( z ) a ( z ) y ] a ( z ) D ( z ) a ( z ) y ] a ( z ) , (A1) a ( z ) = 0 . z ) − . + 1 . , (A2) airwise velocity between dark matter halos a ( z ) = 0 . z . + 0 . , (A3) a ( z ) = − .
710 ( z − . + 5 . , (A4) y ≡ b L ( M , z ) + b L ( M , z ) , (A5) B ρ ( M , M , z ) = (26 . z + 2 . z + 17 . (cid:18) M + M h − M (cid:12) (cid:19) . , (A6) C ρ ( M , M , z ) = − (cid:2) .
109 ( z − . + 0 . (cid:3) (cid:18) M + M h − M (cid:12) (cid:19) [ − . z − . +0 . ] , (A7)where D ( z ) is the linear growth factor normalized to unity at z = 0, and b L ( M, z ) is the linear halo bias.For Eq. (24), we adopt the following forms of C (0) t ( M , M , z ) = (cid:2) − .
36 ( z − . + 60 . (cid:3) ( R , ) . z − . z +1 . , (A8) C (1) t ( M , M , z ) = (cid:2) − .
36 ( z − . + 7 . (cid:3) ( R , ) . z − . z +0 . , (A9) C (2) t ( M , M , z ) = 0 . , (A10) p t ( M , M , z ) = (cid:2) .
866 ( z − . − . (cid:3) ( R , ) .
333 (1+ z ) − . − . , (A11) q t ( M , M , z ) = − . , (A12) C (0) r ( M , M , z ) = (cid:2) . z − z + 2970 (cid:3) exp (cid:34) − (cid:18) . z − . z + 1 . R , (cid:19) (cid:35) R , larger , (A13) C (1) r ( M , M , z ) = (cid:2) − . z + 0 . z + 26 . (cid:3) ( R , ) − . z − . z +0 . , (A14) C (2) r ( M , M , z ) = (cid:2) − .
109 ( z − . + 0 . (cid:3) ( R , ) − . − . z , (A15) p r ( M , M , z ) = − . , (A16) q r ( M , M , z ) = − . , (A17)where R , = R ( M ) + R ( M ) and R , larger = MAX ( R ( M ) , R ( M )). The radii R , and R , larger are in the unit of comoving h − Mpc. B. HALO-BASED STREAMING MODEL OF REDSHIFT-SPACE CLUSTERING WITH A HALOOCCUPATION DISTRIBUTIONIn this appendix, we briefly summarize an analytic expression of the redshift-space two point correlation with ourmodel of the pairwise velocity distribution of dark matter halos for a given halo occupation distribution (HOD) (alsosee Tinker 2007, for more details). Note that we omit the redshift z for most parts in the following discussion forsimplicity. B.1. Setup
For a galaxy sample of interest, we assume that the galaxies can be decomposed into two types, centrals and satellites.For the central galaxies, we assume that they reside in the center of their host dark matter halos and individual hosthalos can have single central galaxies at most. For the satellite galaxies, we populate satellite galaxies to a halo onlywhen a central galaxy exists. The HOD represents the mean number of galaxies in host halos with mass M and it isgiven by (cid:104) N gal (cid:105) M = (cid:104) N cen (cid:105) M + (cid:104) N sat (cid:105) M , (B18)where (cid:104) N cen (cid:105) M and (cid:104) N sat (cid:105) M are the HODs for centrals and satellites, respectively. In the following, we assume thatthe conditional distribution of the number of central galaxies in a given halo follows the Bernoulli distribution (i.e.,can take only zero or one) with mean of (cid:104) N cen (cid:105) M . On the other hand, the conditional distribution of the number ofsatellites is set by the Poisson distribution with mean λ M . In this setup, the HOD for satellites can be expressed as (cid:104) N sat (cid:105) M = (cid:104) N cen (cid:105) M λ M . Once the HOD is specified, we can compute the mean number density of the galaxies as¯ n g = (cid:90) d M d n d M ( (cid:104) N cen (cid:105) M + (cid:104) N sat (cid:105) M ) , (B19)4 Shirasaki et al. where d n/ d M is the halo mass function. In this paper, we adopt the model of halo mass functions in Tinker et al.(2008). B.2. Two-point correlation function
Within the HOD framework, the two-point correlation function of galaxies can be decomposed into two parts knownas one-halo and two-halo terms. The one-halo term represents the two-point correlation within single halos, whiletwo-halo term arises from the clustering among neighboring halos. For a given HOD in Appendix B.1, the one-haloterms in redshift space can be expressed as (Tinker 2007), ξ S ( s p , s π ) = ξ S, cs1h ( s p , s π ) + ξ S, ss1h ( s p , s π ) , (B20) ξ S, cs1h ( s p , s π ) = 12 π ¯ n (cid:90) d M d n d M (cid:104) N sat (cid:105) M (cid:90) ∞−∞ H ( z ) d r π (1 + z ) F cs (cid:16)(cid:113) s p + r π | M (cid:17) s p + r π P cs (cid:18) v z = H ( z )( s π − r π )(1 + z ) (cid:12)(cid:12)(cid:12) M (cid:19) , (B21) ξ S, ss1h ( s p , s π ) = 12 π ¯ n (cid:90) d M d n d M (cid:104) N cen (cid:105) λ M (cid:90) ∞−∞ H ( z ) d r π (1 + z ) F ss (cid:16)(cid:113) s p + r π | M (cid:17) s p + r π P ss (cid:18) v z = H ( z )( s π − r π )(1 + z ) (cid:12)(cid:12)(cid:12) M (cid:19) , (B22)where F cs ( r | M ) is the fraction of number of central-satellite pairs at the radius of r in a halo with M , F ss ( r | M ) is thefraction of number of satellite-satellite pairs, P cs and P ss are the PDF of the pairwise velocity along a line of sight forcentral-satellite and satellite-satellite pairs, respectively. Note that (cid:82) d r F cs ( r | M ) = (cid:82) d r F ss ( r | M ) = 1.When assuming the velocity distribution within each halo as an isotropic, isothermal Gaussian distribution and thesatellite galaxy velocity dispersion in a halo is set to the virial dispersion, one can find P cs ( v z | M ) = N ( v z , , σ vir ,M ) , (B23) P ss ( v z | M ) = N ( v z , , √ σ vir ,M ) , (B24)where N ( x, µ, σ ) is a Gaussian distribution of a random field x with mean µ and variance σ , and σ vir ,M representsthe virival dispersion in a halo with mass M .In addition, it is commonly assumed that the number density profile of satellites follows the mass density profileof its host dark matter halos. When the mass density profile in a halo is described by the (truncated) NFW profile(Navarro et al. 1996), the fraction of number of galaxy pairs is given by F cs ( r | M ) = 1 f ( c ) r ( r + r s ) , (B25) F ss ( r | M ) = r f ( c ) r s (cid:90) c d x Q ( x , r/r s , c ) , (B26)where c and r s are the halo concentration and scaled radius for the NFW profile, f ( c ) = ln(1 + c ) − c/ (1 + c ), and Q ( x , x, c ) = | x − x | > c )(1 + x ) − (cid:2) (1 + | x − x | ) − − (1 + x + x ) − (cid:3) ( x + x < c , | x − x | ≤ c )(1 + x ) − (cid:2) (1 + | x − x | ) − − (1 + c ) − (cid:3) ( x + x ≥ c , | x − x | ≤ c ) . (B27)The two-halo term is then modeled by1 + ξ S ( s p , s π ) = (cid:90) ∞−∞ H ( z ) d r π (1 + z ) P , g (cid:18) v z = H ( z )( s π − r π )(1 + z ) (cid:12)(cid:12)(cid:12) s p , r π (cid:19) (cid:104) ξ ( (cid:113) s p + r π ) (cid:105) , (B28)where ξ ( r ) is the two-halo term of real-space correlation function, and P , g is the pairwise velocity PDF of galaxiesfor two separated halos. We here suppose that ξ ( r ) is accurately predicted by some approach such as perturbation-theory-based models (e.g. Desjacques et al. 2018, for a recent review), semi-analytic models (e.g. Hamana et al. 2001;Tinker et al. 2005; van den Bosch et al. 2013), and simulation-based models (e.g. Kwan et al. 2015; Nishimichi et al.2019; Zhai et al. 2019). For the pairwise velocity PDF, we first compute the pairwise velocity PDF of dark matterhalos for the line-of-sight component by using Eq. (7): P ( v z | r p , r π , M , M ) = (cid:90) d v t P ( v r , v t | r, M , M ) δ D (cid:18) v t − v r cos θ − v z sin θ (cid:19) = (cid:90) d δ N (cid:18) v z , µ r [ δ ] cos θ, (cid:113) Σ r [ δ ] cos θ + Σ t [ δ ] sin θ (cid:19) F ( δ | r, M , M ) , (B29) airwise velocity between dark matter halos r = (cid:113) r p + r π , cos θ = r π /r , F is the condtional PDF of cosmic mass density having a halo pair with massesof M and M within r , µ r is the mean radial velocity at a given environmental density δ , and Σ t,r represents thevelocity dispersion at a given δ . The details of these functions are found in Section 2 and Appendix A. We thenincorporate Eq. (B29) with the HOD framework by assuming the Gaussian velocity distribution of satellites with thevirial dispersion of σ vir . The final expression of P , g is given by P , g ( v z | r p , r π ) = (cid:0) n (cid:48) g (cid:1) − (cid:90) M lim , M min , d M d n d M (cid:104) N gal (cid:105) M (cid:90) M lim , M min , d M d n d M (cid:104) N gal (cid:105) M × P g+h ( v z | r p , r π , M , M ) , (B30) P g+h ( v z | r p , r π , M , M ) = (cid:90) d δ (cid:88) i =1 w i N ( v z , µ r cos θ, σ i ) F ( δ | r, M , M ) , (B31) (cid:0) n (cid:48) g (cid:1) = (cid:90) M lim , M min , d M d n d M (cid:104) N gal (cid:105) M (cid:90) M lim , M min , d M d n d M (cid:104) N gal (cid:105) M , (B32)where M min , is the minimum halo mass that can host a galaxy (usually set by a sufficient small value), and w = (cid:104) N cen (cid:105) M (cid:104) N cen (cid:105) M (cid:104) N gal (cid:105) M (cid:104) N gal (cid:105) M , w = (cid:104) N cen (cid:105) M (cid:104) N sat (cid:105) M (cid:104) N gal (cid:105) M (cid:104) N gal (cid:105) M , w = (cid:104) N sat (cid:105) M (cid:104) N cen (cid:105) M (cid:104) N gal (cid:105) M (cid:104) N gal (cid:105) M , w = (cid:104) N sat (cid:105) M (cid:104) N sat (cid:105) M (cid:104) N gal (cid:105) M (cid:104) N gal (cid:105) M , (B33) σ = Σ r ( δ, r, M , M ) cos θ + Σ t ( δ, r, M , M ) sin θ, (B34) σ = σ + σ ,M , σ = σ + σ ,M , σ = σ + σ ,M + σ ,M . (B35)In Eqs. (B30) and (B32), we set the upper limits of the integral to R ( M lim , ) = r − R ( M min , ) and R ( M lim , ) = r − R ( M ) by taking into account the effect of halo exclusion. C. PERFORMANCE EVALUATION OF OUR MODEL FOR PAIRWISE VELOCITY DISTRIBUTION OFDARK MATTER HALOSIn this appendix, we evaluate our model precision for the profiles of the mean and dispersion in the pairwise velocityof dark matter halos in a wide range of halo masses and redshifts. Figures 13-16 summarize the ratio of the velocitymoments between the simulation results and our model predictions for different halo masses ( M ≥ M ) and redshifts.In each figure, the three left panels show the results for M = 10 . − h − M (cid:12) . From top to bottom, each panel showsthe ratio of (cid:104) v r (cid:105) , σ t , and σ r , respectively. The three middle panels present the results for M = 10 − . h − M (cid:12) ,while the three right panels are for M = 10 . − h − M (cid:12) . Figures 13, 14, 15, and 16 provide the results at z = 0,0.30, 0.55, and 1.01, respectively. There are 10-20%-level differences for halo masses greater than ∼ . h − M (cid:12) ,but our model can reproduce the simulation results for 10 . < M [ h − M (cid:12) ] < . at 0 . < ∼ z < z < Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013, ApJ,762, 109, doi: 10.1088/0004-637X/762/2/109Bianchi, D., Percival, W. J., & Bel, J. 2016, MNRAS, 463,3783, doi: 10.1093/mnras/stw2243Bond, J. R., & Myers, S. T. 1996, ApJS, 103, 1,doi: 10.1086/192267Coles, P., & Jones, B. 1991, MNRAS, 248, 1,doi: 10.1093/mnras/248.1.1Cooray, A. 2006, MNRAS, 365, 842,doi: 10.1111/j.1365-2966.2005.09747.x Cooray, A., & Sheth, R. 2002, PhR, 372, 1,doi: 10.1016/S0370-1573(02)00276-4Crocce, M., Pueblas, S., & Scoccimarro, R. 2006, MNRAS,373, 369, doi: 10.1111/j.1365-2966.2006.11040.xCroton, D. J., Gao, L., & White, S. D. M. 2007, MNRAS,374, 1303, doi: 10.1111/j.1365-2966.2006.11230.xCuesta-Lazaro, C., Li, B., Eggemeier, A., et al. 2020, arXive-prints, arXiv:2002.02683.https://arxiv.org/abs/2002.02683De Bernardis, F., Aiola, S., Vavagiakis, E. M., et al. 2017,JCAP, 2017, 008, doi: 10.1088/1475-7516/2017/03/008 Shirasaki et al. . . . h v r i S i m / h v r i M o d e l . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < . . . σ t , S i m / σ t , M o d e l . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < r [ h − Mpc]0 . . . σ r , S i m / σ r , M o d e l r [ h − Mpc] 0 10 20 30 40 r [ h − Mpc] z = 0 . Figure 13.
The mean and dispersion of pairwise-velocity of dark matter halos with various masses at the redshift of z = 0.Each panel shows the ratio of the velocity-moment profiles for different halo masses. The three left panels show the ratio of (cid:104) v r (cid:105) , σ t , and σ r for the halo masses of M = 10 . − h − M (cid:12) and M ≥ M from top to bottom. The middle and right panelsrepresent the results for M = 10 − . h − M (cid:12) and M = 10 . − h − M (cid:12) , respectively. For a reference, the gray filled regionin each panel shows ± airwise velocity between dark matter halos . . . h v r i S i m / h v r i M o d e l . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < . . . σ t , S i m / σ t , M o d e l . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < r [ h − Mpc]0 . . . σ r , S i m / σ r , M o d e l r [ h − Mpc] 0 10 20 30 40 r [ h − Mpc] z = 0 . Figure 14.
Similar to Figure 13, but this figure presents the results at the redshift of z = 0 . . . . h v r i S i m / h v r i M o d e l . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < . . . σ t , S i m / σ t , M o d e l . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < r [ h − Mpc]0 . . . σ r , S i m / σ r , M o d e l r [ h − Mpc] 0 10 20 30 40 r [ h − Mpc] z = 0 . Figure 15.
Similar to Figure 13, but this figure presents the results at the redshift of z = 0 . Shirasaki et al. . . . h v r i S i m / h v r i M o d e l . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < . . . σ t , S i m / σ t , M o d e l . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . . ≤ M [ h − M (cid:12) ] < ≤ M [ h − M (cid:12) ] < . r [ h − Mpc]0 . . . σ r , S i m / σ r , M o d e l r [ h − Mpc] 0 10 20 30 40 r [ h − Mpc] z = 1 . Figure 16.
Similar to Figure 13, but this figure presents the results at the redshift of z = 1 . airwise velocity between dark matter halos29