Redshift-Space Galaxy Clustering: Accurate Modeling, Velocity Bias, and Assembly Effect
Zheng Zheng University of Utah
Hong Guo (SHAO), Jia-Ni Ye (SHAO), Idit Zehavi (CWRU), Kevin McCarthy (Utah), Xiaoju Xu (Utah) KITP 2017 Quantifying and Understanding the Galaxy-Halo Connection 5/16/17 Redshift-Space Galaxy Clustering
Small scales: FoG galaxy kinematics inside virtualized structures (halos)
Large scales: Kaiser effect structure growth rate
The simple model is not accurate. High Precision Galaxy Clustering Measurements High Precision Galaxy Clustering Measurements
~ accuracy of analytic models of real-space 2PCFs (e.g., Tinker+05, van den Bosch+13) Difficulties in Developing Accurate Models of Galaxy Clustering
• non-linear evolution of matter power spectrum • scale dependence of halo bias • halo exclusion effect • nonsphericity of halos • halo alignment • …
(Zheng04, Tinker+05, van den Bosch+13) Difficulties in Developing Accurate Models of Galaxy Clustering Galaxy Infall onto SDSS Groups 5 P (v ,v r, M ,M ) r t| 1 2 Distribution of halo-halo (radial and transverse) pairwise velocity
(e.g., Tinker 2007, Reid & White 2011, Zu & Weinberg 2013)
Figure 3. Joint probability distributions of radial and tangential velocities P (vr,vt) from the simulation (top panels) and the best–fit using our GIKmodel(bot- Reid & Whitetom panels), in four(2011) different radial bins marked at the bottom of each panel. The colourZu scales & Weinberg used by panels in the same (2013) column are identical, indicated by the colour bar on top.
−1 T −1 P (vr,vt) across all scales from the inner 1 h Mpc to beyond element vectors, and Qv =(v − ¯v ) Σ (v − ¯v ) is a scalar where 40 h−1Mpc. σ2 0 Motivated by the top panels of Fig. 3, we adopt a two– Σ = rad . (5) 0 σ2 component mixture model for the velocity distribution at anygiven " tan# cluster–centric radius r,withthevirializedcomponentdescribed For the two rhs terms in Equation 4, t2 is the density function of by a 2D Gaussian G and the infall component by a 2D skewed t- 2D t-variate with dof degrees of freedom, distribution T : (dof+2)/2 Γ{(dof +2)/2} Q t (v; dof)= 1+ v , 2 |Σ|1/2(π dof)1/2Γ(dof/2) dof P (vr,vt) ≡ P (v)=fvir ·G(v)+(1− fvir) ·T(v), (3) " # (6) dof where f ! 0 is the fraction of galaxies in the virialized com- and T1(x; +2)denotes the scalar t-distribution function with vir dof α ponent, approaching zero at large r.Werefertotheradiusbeyond +2degrees of freedom. Generally speaking, controls the dof which f =0as the “shock radius” r ,sinceitmarks(atleast skewness of P (vr,vt) in the radial velocity direction, while vir sh dof within the model) the boundary between single–component and adjusts the kurtosis in both directions, with lower correspond- two–component flow. By definition G has zero mean in both ra- ing to longer non–Gaussian tails. Since P (vr,vt) is symmetric α dial and tangential axes, and we find it adequate to assume equal in the tangential velocity axis, is reduced to one parameter α. dispersions, making G afunctionofonlyoneparameter,thevirial σrad and σtan describe the dispersion in each direction, and vr,c is the characteristic radial velocity. Therefore, we have seven pa- dispersion σvir (which is still allowed to vary with r). For the in- fall component, describing the varying degrees of skewness and rameters in total for P (vr,vt) at every r:virializedfractionfvir, kurtosis at different r requires a functional form T with greater velocity dispersion of the virialized component σvir,characteristic complexity. We adopt the skewed t–distribution parameterization infall velocity vr,c,twovelocitydispersionsoftheinfallcompo- from Azzalini & Capitanio (2003), with two parameters describing nent σrad and σtan,skewnessparameterα,andkurtosisparameter dof the higher order moments of the velocity distribution (α and dof) (effectively reducing to five parameters at r>rsh). With seven parameters, Equation 3 provides an excellent fit for the measured in addition to three parameters for the mean and dispersions (vr,c , P (vr,vt) at all scales, as shown visually in the bottom panels of σrad,andσtan). The full expression is Fig. 3, and in greater detail below. We considered other parameteri- T (v)=2t2(v; dof) × zations for the infall component, such as sums of Gaussians, but we dof were unable to find a compact description as accurate as the skewed αT ω−1 v ¯v +2 dof T1 ( − ) · ; +2 (4), t-distribution, so we obtained poor results in modeling ξs . Qv + dof cg ! " # $ Using the best–fit GIK models, we take a closer look into the where ¯v = vr,c , 0 , α = α, 0 ,andω = σrad, σtan are 2- properties of P (vr,vt) at different radii in Fig. 4. In each panel, % & % & % & ⃝c 0000 RAS, MNRAS 000,000–000 Model Galaxy Clustering with N-body Simulations
Populate halos with galaxies according to HOD/CLF to form mock Measure 2PCFs from the mock as the model prediction
e.g., White+(2011), Parejko+(2013) Credit: Springel+(2005) halotools (Hearin+2016) 2 Zheng Zheng and Hong Guo halo galaxy pairs. The intra-halo component, or the one-haloterm, 2SIMULATION-BASEDMETHODOFCALCULATING represents the highly nonlinear part of the 2PCF. The inter-halo GALAXY 2PCFS component, or the two-halo term, can be largely modelled by linear In our simulation-based method, we divide haloes identified in N- theory. Such analytic models have the advantage of being compu- body simulations into narrow bins of a given property, which de- tationally inexpensive, and they can be used to efficiently probe the termines galaxy occupancy. In the commonly used HOD/CLF, the HOD/CLF and cosmology parameter space. However, as the preci- property is the halo mass. In our presentation, we use halo mass as sion of the 2PCF measurements in large galaxy surveys continues the halo variable, but the method can be generalized to any setof to improve, the requirement on the accuracy of the analytic mod- halo properties. els becomes more and more demanding. As pointed out in Zheng The basic idea of the method is to decompose the galaxy (2004a), an accurate model of the galaxy 2PCF needs to incorporate 2PCF into contributions from haloes of different masses, from one- the nonlinear growth of the matter power spectrum (e.g. Smith et al. halo and two-halo terms, and from different types of galaxy pairs 2003), the halo exclusion effect, and the scale-dependent halo bias. (e.g. central-central, central-satellite, and satellite-satellite pairs). In addition, the non-spherical shape of haloes should also beac- The decomposition also allows the separation between the halo counted for (e.g. Tinker et al. 2005; van den Bosch et al. 2013). occupation and halo clustering. The former relies on the specific These are just factors to be taken into account in computing the HOD/CLF parameterization, while the latter can be calculated from real-space or projected 2PCFs. For redshift-space 2PCFs, more fac- the simulation. The method is to tabulate all relevant information tors come into play. An accurate analytical description of the ve- about the latter for efficient calculation of galaxy 2PCFs andexplo- locity field of dark matter haloes in the nonlinear or weakly non- ration of the HOD/CLF parameter space. linear regime proves to be difficult and complex (e.g. Tinker 2007; We first formulate the method in the HOD/CLF framework. Reid & White 2011; Zu & Weinberg 2013). Therefore, an accurate We then apply the similar idea to the SHAM case, which provides analytic model of redshift-space 2PCFs on small and intermediate amoregeneralSHAMmethod. scales is still not within reach.
The above complications faced by analytic models can all be avoided or greatly reduced if the 2PCF calculation is directly done 2.1 Case with Simulation Particles with the outputs of N-body simulations. With the simulation, dark Let us start with a given N-body simulation and a given set of matter haloes can be identified, and their properties (mass, veloc- HOD/CLF parameters. To populate galaxies into a halo identified in ity, etc) can be obtained. For a given set of HOD/CLF parameters, the simulation, we can put one galaxy at the halo ‘centre’ as a cen- one can populate haloes with galaxies accordingly (e.g. using dark tral galaxy, according to the probability specified by the HOD/CLF matter particles as tracers) and form a mock galaxy catalog. The parameters. Halo ‘centre’ should be defined to reflect galaxy for- 2PCFs measured from the mock catalog are then the model pre- mation physics. For example, a sensible choice is the position of dictions used to model the measurements from observations. Such potential minimum rather than centre of mass. For satellites, we amethodofdirectlypopulatingsimulationshavebeendeveloped can choose particles as tracers. In the usually adopted models, it is and applied to model galaxy clustering data (e.g. White et al. 2011; assumed that satellite galaxies follow dark matter particles inside Parejko et al. 2013). This simulation-based model is attractive, as haloes (e.g. Zheng 2004a; Tinker et al. 2005; van den Bosch et al. more and more large high-resolution N-body simulations emerge. 2013), rooted in theoretical basis (e.g. Nagai & Kravtsov 2005). It is also straightforward to implement. Once the mock catalog is One can certainly modify the distribution profile as needed, and produced, measuring the 2PCFs can be made fast (e.g. with tree below we assume that the distribution of galaxies inside haloes has code). However, populating haloes with a given set of HOD/CLF been specified and that the corresponding tracer particlesGalaxy have been Clustering Modelling with Simulations 3 parameters is probably the most time-consuming step, as one needs selected for each halo. to loop over all haloes of interest. In addition, informationofindi- The right-handWe side divide (RHS) haloes is the same in the quantity simulation from counting into N one-narrow massnss− binspair,2h,wereachthefinalexpressionforthetwo-haloterm, vidual haloes and tracer particles is needed, like their positions andhalo pairs in each halo and the summation is over all the halo mass and denote the mean number density of haloes in the mass2h bin n¯in¯j Morebins. Efficient Here ⟨Npair Simulation-Based(M)⟩ is the total mean number Clustering of galaxy pairs in Modelingr r velocities. Even with only a subset of all the particles in a high- ξgg ( )= 2 ⟨Ncen(Mi)⟩⟨Ncen(Mj )⟩ξhh,cc( ; Mi,Mj ) log Mi ± d log Mri/2 as n¯i.Themeannumberdensityofgalax- n¯g resolution simulation, the amount of data can still be substantial. haloes of mass M,andf( ; M) is the probability distribution of !i̸=j pair separationies is computed in haloes of as mass M,i.e.f(r; M)d3r is the proba- HOD Halo Properties n¯in¯j r bility of finding pairs with separation in the range r±dr/2 in haloes + 2 2 ⟨Ncen(Mi)⟩⟨Nsat(Mj )⟩ξhh,cs( ; Mi,Mj ) The purpose of this paper is to introduce a method that takes n¯g of M.Byfurtherdecomposingpairsintocentral-satellite(cenn¯g = [⟨Ncen(Mi)⟩ + ⟨Nsat(Mi)⟩]¯ni, -sat)Mass Function (1)!i̸=j the advantage of the simulation-based model, but being much more and satellite-satellite!i (sat-sat) pairs, we reach the following expres- n¯in¯j r + 2 ⟨Nsat(Mi)⟩⟨Nsat(Mj )⟩ξhh,ss( ; Mi,Mj ) (9) efficient in modelling galaxy clustering. The main idea is to decom-sion, n¯g pose the galaxy 2PCFs and compress the information in the simu- where Ncen(M) and Nsat(M) are the occupation numbers of cen- !i̸=j 1htralr and satelliten¯i galaxies in a halo of massr , denotes the aver- lation by tabulating relevant clustering-related quantities of dark1+ξgg ( )= 2 2 ⟨Ncen(Mi)Nsat(Mi)⟩fcs( ; MiM) ⟨⟩ Equations (1), (3), and (9)leadtothemethodwepro- n¯g matter haloes. We also apply a similar idea to extend the com- age over!i all haloes of this mass, and i =1,...,N. Profilepose. The quantities related to galaxy occupancy are specified Inn¯ thei halo-based model, galaxy 2PCF ξgg is computed as monly used sub-halo abundance matching method (SHAM; e.g. Galaxy+ Clustering⟨Nsat(M Modelling)[Nsat(M ) with− 1]⟩f Simulationsss(r; M ) (3)3 by the HOD/CLF parameterization one chooses, while those re- 2 i i i 1h 2h n¯g lated to haloes are from the simulation, independent of the Conroy et al. 2006). the!i combination of two terms, ξgg =1+ξgg + ξgg (Zheng 1h 2h The right-hand side (RHS) is the same quantity from counting one- nss2004a−pair,2h),,wereachthefinalexpressionforthetwo-haloterm, where the one-halo term ξgg (two-halo term ξgg )arefromHOD/CLF parameterization. We therefore can prepare tables for The functions fcs(r; M) and fss(r; M) are the probability distri- r r r r halo pairs in each halo and the summation is over all the halo mass contributions of intra-halo (inter-halo) galaxy pairs. Fon¯llowingi, fcs( ; Mi), fss( ; Mi), ξhh,cc( ; Mi,Mj ), ξhh,cs( ; Mi,Mj ), The paper is structured as follows. In Section 2,weformu-butions2h of one-halon¯i cen-satn¯j and sat-sat galaxy pair separation in bins. Here ⟨Npair(M)⟩ is the total mean number of galaxy pairs in r r and r .ForagivensetofHOD/CLFparameters, ξgg ( )= 2 ⟨Ncen(Mi)⟩⟨Ncen(Mj )⟩ξhh,cc( ; Mi,Mj ) ξhh,ss( ; Mi,Mj ) late the method, within the HOD/CLF-liker framework and withinhaloes ofBerlind mass M &.Theyarenormalizedsuchthat Weinbergn¯g (2002), the one-halo term can be computed haloes of mass M,andf( ; M) is the probability distribution of !i̸=j the predictions of galaxy 2PCFs can be obtained from perform- 3 based on the halo/sub-halopair separation framework. in haloes of In mass SectionM,i.e.3,weshowanexamplef(r; M)d r is the proba- n¯ n¯ ing the weighted summation over the tables. The tables are only 3 i j 3 r bility of finding pairs with separation in the range r±dr/2 in haloes fcs(r; M+)d r =12 2 and⟨Ncen(Mfssi)(⟩⟨r;NMsat)d(Mrj )=1⟩ξhh. ,cs( ; Mi,Mj(4)) Clustering of modelling redshift-space 2PCFs, which also provides an under- 1 3 n¯g 1h prepared3 once, and we can then change the galaxy occupation as " n¯ (¯!ni̸=jd r) 1+"ξ (r) = n¯ ⟨Npair(M )⟩f(r; M )d r.(2) standing ofM the.Byfurtherdecomposingpairsintocentral-satellite(cen three-dimensional (3D) small- and intermed-sat)iate- 2 g g gg i i ineeded to compute galaxy 2PCFs for different galaxy samples and and satellite-satellite (sat-sat) pairs, we reach the following expres- Note that here andn in¯in¯ whatj follows, the 2PCF!i can ber either real- scale galaxy redshift-space 2PCF and its multipoles by decompos- + 2" ⟨Nsat(Mi)⟩⟨#Nsat(Mj )⟩ξhh,ss( ; Mi,Mj ) (9) different sets of HOD/CLF parameters, which is much more effi- sion, n¯g space, projected-space,!i̸=j redshift-space, or it can be the multipoles cient than populating galaxies into haloes by selecting particles. ing them into the various components. In Section 4,wesummarize The left-hand side (LHS) is ther number density of one-halo pairs 1h r n¯i r of the redshift-space 2PCF. The variabler shouldr be understood as Since summation is used to replace integration in the method, the method1+ andξgg discuss( )= possible2 2 ⟨Ncen generalizations(Mi)Nsat(Mi)⟩f andcs( limita; Mi) tions. withEquations separation (1), ( in3), the and range (9)leadtothemethodwepro-± d /2 from the definition of 2PCF. n¯g pair separation in the corresponding space. For redshift-space clus- we need to choose narrow halo mass bins (d log M =0.01 is usu- !i pose. The quantities related to galaxy occupancy are specified n¯ tering, we discuss how to specify velocity distribution of galaxies ally sufficient, as shown in Section 3). The n¯i table represents the i r by the HOD/CLF parameterization one chooses,c while those re- + 2 ⟨Nsat(Mi)[Nsat(Mi) − 1]⟩fss( ; Mi) (3) later. ⃝ 0000 RAS, MNRAS 000,000–000halo mass function. To prepare the other tables that depend on n¯g lated to haloes are from the simulation, independent of the !i To compute the two-halo term, weadd up all possible two-halo pair separation, the bins of pair separation r are best chosen to HOD/CLF parameterization. We therefore can prepare tables for The functions fcs(r; M) and fss(r; M) are the probability distri- galaxy pairs, following the 2PCF decomposition from different pair match the ones used in the measurements from observational data, n¯ , fcs(r; M ), fss(r; M ), ξ (r; M ,M ), ξ (r; M ,M ), countsi in Zu et al.i (2008). Similari hh to,cc equationi (j2), thehh,cs total numberi j butions of one-halo cen-sat and sat-sat galaxy pair separation in and ξ (r; M ,M ).ForagivensetofHOD/CLFparameters, which would naturally avoid any discrepancy related to the finite density ofhh two-halo,ss pairsi j with separation in the range r r is haloes of mass M.Theyarenormalizedsuchthat the predictions of galaxy 2PCFs can be obtained from± d perform/2 - bin sizes. For haloes in each mass bin, the fcs and fss tables can ing the weighted summation over the tables. The tables are only be computed by using either all the particles in the haloes with 3 3 1 3r 2h r fcs(r; M)d r =1 and fss(r; M)d r =1. (4) npairprepared,2h = n¯ once,g(¯ng andd ) we1+ canξgg then( ) change, the galaxy occupation(5) as the specified distribution or a random subset. For ξhh,cc, ξhh,cs, " " 2 needed to compute galaxy# 2PCFs$ for different galaxy samples and and ξhh,ss,weeffectivelycomputethehalo-halotwo-pointcross- Note that here and in what follows, the 2PCF can be either real- whichdifferent is composed sets of of HOD/CLF two-halo parameters, central-central which (cen-cen) is much pa moreirs effi- correlation function with different definitions of halo positions. For space, projected-space, redshift-space, or it can be the multipoles cient than populating galaxies into haloes by selecting particles. ξ ,halopositionsaredefinedbyourchoiceof‘centres’.For 1 3r hh,cc of the redshift-space 2PCF. The variable r should be understood as ncc−pair,2h = [¯ni⟨Ncen(Mi)⟩][¯nj ⟨Ncen(Mj )⟩d ] r Since summation2 is used to replace integration in the method, ξhh,cs( ; Mi,Mj ),wechoose‘centres’forMi haloes and positions i̸=j pair separation in the corresponding space. For redshift-space clus- we need to choose! narrow halo mass bins (d log M =0.01 is usu- of arbitrary tracer particles in Mj haloes. For ξhh,ss(r; Mi,Mj ), tering, we discuss how to specify velocity distribution of galaxies r ally sufficient, as×[1 shown + ξhh in,cc Section( ; Mi3,M). Thej )],n¯i table represents(6) the positions of arbitrary tracer particles in both Mi and Mj haloes are later. halo mass function. To prepare the other tables that depend on chosen. We can use any number of tracer particles in each halo to two-halo cen-sat pairs To compute the two-halo term, weadd up all possible two-halo pair separation, the bins of pair separation r are best chosen to do the calculation. For haloes with positions defined by the tracer galaxy pairs, following the 2PCF decomposition from different pair match the ones used in the measurements from observational data, 3r particles, they can be thought as extended (with positions having a counts in Zu et al. (2008). Similar to equation (2), the total number ncs−pairwhich,2h would= naturally[¯ni⟨ avoidNcen( anyMi) discrepancy⟩][¯nj ⟨Nsat( relatedMj )⟩d to] the finite probability distribution). On large scales, ξhh,cc, ξhh,cs,andξhh,ss r r ̸= density of two-halo pairs with separation in the range ± d /2 is bin sizes. For haloes!i j in each mass bin, the fcs and fss tables can are the same, while on small scales, ξhh,cs and ξhh,ss are smoothed be computed by×[1 using + ξ eitherhh,cs(r all; M thei,M particlesj )], in the haloes(7) with 1 3r 2h r version of ξhh,cc.Notethatinanalyticmodelssuchdifferences npair,2h = n¯g(¯ng d ) 1+ξgg ( ) , (5) the specified distribution or a random subset. For ξhh,cc, ξhh,cs, 2 and two-halo sat-sat pairs are usually neglected. In computing the three halo-halo correlation # $ and ξhh,ss,weeffectivelycomputethehalo-halotwo-pointcross- functions, we do not need to construct random catalogs to find out which is composed of two-halo central-central (cen-cen) pairs correlation function with different definitions of halo positions. For 1 3r the pair counts from a uniform distribution – in the volume nss−pairξ ,2h,halopositionsaredefinedbyourchoiceof‘centres’.For= [¯ni⟨Nsat(Mi)⟩][¯nj ⟨Nsat(Mj )⟩d ] Vsim 1 3r hh,cc 2 ncc−pair,2h = [¯ni⟨Ncen(Mi)⟩][¯nj ⟨Ncen(Mj )⟩d ] r of the simulation with periodic boundary conditions, the counts of 2 ξhh,cs( ; Mi,Mj )!i,wechoose‘centres’for̸=j Mi haloes and positions !i̸=j r cross-pairs at separation in the range r ± dr/2 between two ran- of arbitrary tracer×[1 particles + ξhh ss in(rM; Mj haloes.,M )]. For ξhh,ss( ; Mi,M(8)j ), r , i j ×[1 + ξhh,cc( ; Mi,Mj )], (6) positions of arbitrary tracer particles in both Mi and Mj haloes are domly distributed populations with number densities n¯i and n¯j are 3 In eachchosen. of equations We can use (6)–( any8), number the summation of tracer particles is over all in each halo halo massto simply (¯niVsim)(¯nj d r).Makinguseofthisfactcangreatlyre- two-halo cen-sat pairs bins (i.e.do thei =1 calculation.,...,N and Forj haloes=1,..., withN). positions The three defined correlation by the func- tracer duce the computational expense in preparing the tables. 3 tionsparticles, on the RHS they have can the be thought following as extended meanings (with – ξhh positions,cc(r; M hi,Mavingj ) a For the redshift-space tables, in addition to the halo veloci- ncs−pair,2h = [¯ni⟨Ncen(Mi)⟩][¯nj ⟨Nsat(Mj )⟩d r] is justprobability the two-point distribution). cross-correlation On large scales, functionξhh,cc between, ξhh,cs,and ‘centres’ξhh,ss ties, one needs to specify the velocity distribution of galaxies in- !i̸=j (positionsare the to same, put while central on galaxies) small scales, ofξ haloeshh,cs and ofξ masseshh,ss areM smoothedi and side haloes, which can be different from that of dark matter parti- ×[1 + ξhh,cs(r; Mi,Mj )], (7) Mj (cen-cen);version ofξξhhhh,cs,cc(.Notethatinanalyticmodelssuchdifferencesr; Mi,Mj ) is the two-point cross-correlation cles (a.k.a. velocity bias; e.g. Berlind & Weinberg 2002). The dif- and two-halo sat-sat pairs functionare usually between neglected. the ‘centres’ In computing of Mi haloes the three and halo-halo the satellite correlation tracer ference can be parameterized by central and satellite velocity bias functions, we do not need to construct random catalogs to find out particles in the (extended) Mj haloes (cen-sat); ξhh,ss(r; Mi,Mj ) parameters (e.g. Guo et al. 2015a). For a set of central and satel- 1 3r the pair counts from a uniform distribution – in the volume nss−pair,2h = [¯ni⟨Nsat(Mi)⟩][¯nj ⟨Nsat(Mj )⟩d ] is the two-point cross-correlation function between satellite tracerVsim lite velocity bias parameters and with a choice of the line-of-sight 2 of the simulation with periodic boundary conditions, the counts of !i̸=j particles in the (extended) Mi haloes and those in the (extended) direction, we can obtain the redshift-space positions of thecen- cross-pairs at separation in the range r ± dr/2 between two ran- ×[1 + ξhh,ss(r; Mi,Mj )]. (8) Mj haloes (sat-sat). With npair,2h = ncc−pair,2h + ncs−pair,2h + tral galaxy and satellite tracer particles according to halovelocities domly distributed populations with number densities n¯i and n¯j are 3 In each of equations (6)–(8), the summation is over all halo mass simply (¯niVsim)(¯nj d r).Makinguseofthisfactcangreatlyre- ⃝c 0000 RAS, MNRAS 000,000–000 bins (i.e. i =1,...,N and j =1,...,N). The three correlation func- duce the computational expense in preparing the tables. tions on the RHS have the following meanings – ξhh,cc(r; Mi,Mj ) For the redshift-space tables, in addition to the halo veloci- is just the two-point cross-correlation function between ‘centres’ ties, one needs to specify the velocity distribution of galaxies in- (positions to put central galaxies) of haloes of masses Mi and side haloes, which can be different from that of dark matter parti- Mj (cen-cen); ξhh,cs(r; Mi,Mj ) is the two-point cross-correlation cles (a.k.a. velocity bias; e.g. Berlind & Weinberg 2002). The dif- function between the ‘centres’ of Mi haloes and the satellite tracer ference can be parameterized by central and satellite velocity bias particles in the (extended) Mj haloes (cen-sat); ξhh,ss(r; Mi,Mj ) parameters (e.g. Guo et al. 2015a). For a set of central and satel- is the two-point cross-correlation function between satellite tracer lite velocity bias parameters and with a choice of the line-of-sight particles in the (extended) Mi haloes and those in the (extended) direction, we can obtain the redshift-space positions of thecen- Mj haloes (sat-sat). With npair,2h = ncc−pair,2h + ncs−pair,2h + tral galaxy and satellite tracer particles according to halovelocities
⃝c 0000 RAS, MNRAS 000,000–000 Accurate and Efficient Halo-Based Galaxy Clustering Modeling with Simulations • Accurate - equivalent to populating galaxies into dark matter halos and using the (mean) mock 2PCF measurements as the model prediction - no finite-bin-size effect (same binning and integration scheme as measurements); residual RSD automatically accounted for • Efficient - no need for the construction of mocks and the measurement of the 2PCF from the mocks - independent of simulation size - efficient exploration of the parameter space (e.g., MCMC) Extension to subhalos (SCAM), halo variables other than • mass, and other clustering statistics (Neostein+2011, Neistein & Khochfar2012, Zheng & Guo 2016, Guo+2015) An Accurate and Efficient Simulation-based Model for Redshift-Space Galaxy Two-Point Correlation Function
ZZ & Guo (2016) total one-halo total
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ZZ & Guo (2016) An Accurate and Efficient Simulation-based Model for Redshift-Space Galaxy Two-Point Correlation Function
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ZZ & Guo (2016) An Accurate and Efficient Simulation-based Model for Redshift-Space Galaxy Two-Point Correlation Function
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ZZ & Guo (2016) Modeling Redshift-Space Galaxy Clustering The Astrophysical Journal,762:109(20pp),2013January10 Behroozi, Wechsler, & Wu 6. HALO CORE VELOCITY OFFSETS In the ΛCDM paradigm, the main method by which halos reach a relaxed state is dynamical friction, namely, energy in bulk motions relative to the halo center of gravity is transformed into increased halo velocity dispersion. Dynamical friction depends not only on the background density that a satellite halo • Choose the referenceis passing through, but also frame on the satellite mass and velocity. For a massive host halo, the high background density means that to define galaxyincoming satellitesvelocity will initially transferbias momentum to the host with high efficiency. However, massive host halos also have very strong tidal fields, and once satellites are disrupted into cold velocity streams, dynamical friction becomes much less effective at transferring momentum into the inner halo regions. This combination of dynamical friction and tidal forces suggests - halo core framethat momentum transfer may be more efficient in the outer regions of a host halo and less efficient in the inner regions, leading to an offset between the mean velocity of the central density peak and the bulk velocity of the halo. Behroozi et al. (2013) - halo bulk velocityWe examine frame the presence of this effect by calculating radially averaged halo particle velocities (i.e., particle velocities aver- (more appropriateaged in sphericalfor large, shells) andlow-res comparing tosimulations) the halo bulk velocity (averaged over all halo particles) for a large number of central halos in both the Bolshoi and Consuelo simulations over a range of halo masses (1012–1014 M )andredshifts(z 0–1.4). We additionally consider the effects⊙ of excluding particles= belong- ing to substructure from calculations of the radially binned ve- locities; results for all of the median offsets (both including and excluding substructure) are shown in Figure 11. Figure 11 demonstrates that the core-bulk velocity difference can be quite dramatic at high redshift for massive halos: halos with 1014 M
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BOSS Galaxies z~0.5
Guo, ZZ, et al. (2015a)
Projected Monopole
Quadrupole
Hexadecapole Galaxy Kinematics inside Halos
(( ))
satellite velocity bias satellitevelocity BOSS Galaxies
The central galaxy in a halo central velocity bias is not at rest w.r.t. the halo. Guo, ZZ, et al. (2015a) SDSS Main Galaxy Sample (z~0.1)
Projected Monopole
Quadrupole Hexadecapole
Guo, ZZ, et al. (2015c) SDSS Main Galaxy Sample (z~0.1)
Projected Monopole
Quadrupole Hexadecapole
Guo, ZZ, et al. (2015c) Velocity Bias of SDSS Main Galaxies (z~0.1) Guo, ZZ, et al. (2015c)
faint samples bright samples
In broad agreement with results based on galaxy groups (van den Bosch+2005; Skibba+ 2011) Velocity Bias of SDSS Main Galaxies (z~0.1)
pairwise infall velocity
cen gal velocity dispersion
faint bright
In lower mass halos, central galaxies and halos are more mutually relaxed, consistent with an overall earlier formation and thus more time for relaxation. Velocity Bias in the Illustris Simulation
Ye, Guo, ZZ, & Zehavi (2017) Velocity Bias in the Illustris Simulation Galaxy Velocity Bias in IllustrisYe, Guo, ZZ, & ZehaviGalaxy (2017 Velocity) Bias in 7 Illustris 9
Fig. 6.— Dependence of satellite velocity bias on the time after the satellite galaxies were accreted onto the halos for the center-of-mass cen v bias halo velocity (left) and 10% coresat velocity v bias (right) frames. Lines with di↵erent symbols are for satellite galaxies of di↵erent SHMRs as labeled. more affected by more affected by halo accretion/merger dynamics inside halos
Fig. 4.— Dependence of the central galaxy velocity bias on the halo (top panels) and galaxy (bottom panels) ages for di↵erent SHMRs in di↵erent colors, as labeled. The left panels show our fiducial model of the center-of-mass velocity frame and the right panels display the result for the halo velocity definition with 10% core (see Section 3.2). Fig. 7.— Left: Dependence of ↵s on the distances of the satellite galaxies to the halo centers for the samples of di↵erent SHMRs. Right: ferent masses and for di↵erent SHMRs (solid linesAverage with timesistent after accretion spatial for distributions satellite galaxies with at di the↵erent dark radii matter in the parti- halos. The lines with di↵erent symbols are for samples of di↵erent di↵erent colors). The density profiles are normalizedSHMRs. at cles, while the spatial profile of the satellites with a lower the virial radius Rvir for better comparisons. Thewould black be usefulSHMR to is study much shallower the evolution than of that the of veolicity the dark matter(triangles) in and the dark matter mass in the host halo 13 1 line in each panel is the average density profile forof the central andmassive satellite halos galaxies of Mh > in10 theh halo M frame., as shown Note in the(squares). right The evolution histories of the galaxy veloc- 4 dark matter distributions in these halos. that velocitypanel bias of is Figure defined8. in For a satellites statistical with sense, 10 while
1 ˙ + v =0 (continuity) ar ·
Dawson, et al. (2015) Tightening Cosmological Constraints from Small- and Intermediate-Scale Redshift-Space Distortions
1 ˙ + v =0 (continuity) ar ·
•Probe structure growth rate •Test theories of gravity •Constrain dark energy
Dawson, et al. (2015) Tightening Cosmological Constraints from Small- and Intermediate-Scale Redshift-Space Distortions
d ln D f ⌘ d ln a
large-scale 3D redshift 2PCF
amplitude => b 8
f } => f 8 shape => b insensitive to assembly bias Dawson, et al. (2015) Assembly Effect on Halo Clustering and Kinematics
v12 at 6Mpc/h
Xu & ZZ (in prep) Assembly Effect on Halo Clustering and Kinematics
v12 at 6Mpc/h
Xu & ZZ (in prep) Assembly Effect on fσ8 Constraint from Small Scales
—— w/ assembly bias - - - - w/o assembly bias
McCarthy & ZZ (in prep)
[McEwen & Weinberg (2016) on matter correlation from galaxy correlation function and galaxy lensing] Summary
• accurate and efficient modeling of small- and intermediate-scale redshift-space galaxy clustering by tabulating necessary information of halos in N-body simulations • redshift-space clustering modeling of BOSS CMASS and SDSS Main galaxies to constrain galaxy kinematics inside halos (and tighten fσ8 constraints), with inferred velocity bias in broad agreement with predictions of hydro simulations
• influence of assembly bias on fσ8 constraints and halo assembly on both halo clustering and kinematics