Nonlinear stochastic biasing of halos: Analysis of cosmological N-body simulations and perturbation theories
aa r X i v : . [ a s t r o - ph . C O ] J un Nonlinear stochastic biasing of halos:Analysis of cosmological N -body simulations and perturbation theories Masanori Sato ∗ and Takahiko Matsubara , Department of Physics, Nagoya University, Chikusa, Nagoya 464–8602, Japan and Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,Nagoya University, Chikusa, Nagoya 464–8602, Japan (Dated: October 30, 2018)It is crucial to understand and model a behavior of galaxy biasing for future ambitious galaxy redshift sur-veys. Using 40 large cosmological N -body simulations for a standard Λ CDM cosmology, we study the cross-correlation coe ffi cient between matter and the halo density field, which is an indicator of the stochasticity ofbias, over a wide redshift range 0 ≤ z ≤
3. The cross-correlation coe ffi cient is important to extract informationon the matter density field, e.g., by combining galaxy clustering and galaxy-galaxy lensing measurements. Wecompare the simulation results with integrated perturbation theory (iPT) proposed by one of the present authorsand standard perturbation theory (SPT) combined with a phenomenological model of local bias. The cross-correlation coe ffi cient derived from the iPT agrees with N -body simulation results down to r ∼
15 (10) h − Mpcwithin 0.5 (1.0) % for all redshifts and halo masses we consider. The SPT with local bias does not explaincomplicated behaviors on quasilinear scales at low redshifts, while roughly reproduces the general behavior ofthe cross-correlation coe ffi cient on fully nonlinear scales. The iPT is powerful to predict the cross-correlationcoe ffi cient down to quasilinear regimes with a high precision. PACS numbers: 98.80.Es
I. INTRODUCTION
In the standard cosmological model, known as the Λ CDMmodel, the energy density is dominated by mysterious com-ponents called dark matter and dark energy. The correlationfunction of dark matter and its Fourier counterpart, the powerspectrum, contain a wealth of information that can be usedto determine, e.g., the dark matter, dark energy, and neutrinomasses. Thus, it is very important to exploit these quanti-ties in the large-scale structure of the universe, which is apillar of modern observational cosmology. However, how totake account of galaxy biasing needs to be investigated. Ob-servable galaxies are biased relative to the underlying matterdensity field. The galaxy biasing is a ff ected by nonlinear ef-fects and is scale dependent in general. Such nonlinear e ff ectsimpose a serious problem in analyzing galaxy surveys [e.g.,1–4]. Upcoming galaxy surveys such as BigBOSS [5], Eu-clid [6], Subaru Prime Focus Spectrograph (PFS) [7], and theWide-Field Infrared Survey Telescope (WFIRST) require theunderstanding of galaxy biasing with high precision and thusa theoretically precise description of the galaxy biasing is acrucial issue.Most of the direct studies of the clustering of matter oncosmological scales rely on shear-shear weak lensing, but itis also possible to extract information on the matter cluster-ing by combining galaxy clustering and galaxy-galaxy lensingmeasurements [e.g., 8]. To achieve this, one has to preciselyknow the relation between the distribution of galaxies and thedistribution of matter. An important property of the relation ∗ [email protected] http: // bigboss.lbl.gov / http: // sumire.ipmu.jp / pfs / intro.html http: // wfirst.gsfc.nasa.gov / is often characterized by a cross-correlation coe ffi cient. Thecross-correlation coe ffi cient is a characteristic parameter ofstochasticity [9]. Since galaxies are expected to form in darkmatter halos in modern models of galaxy formation, under-standing and modeling the clustering properties of the halosplay an important role and are crucial first steps in modelinggalaxy biasing.In this work, we examine how well-known models of haloclustering reproduce the cross-correlation coe ffi cient betweenmatter and halo density fields obtained from N -body simu-lations. We consider two models of nonlinear bias: the in-tegrated Perturbation Theory (iPT) developed by Matsubara[10] which naturally incorporates the halo bias, redshift-spacedistortions, nonlocal Lagrangian bias, and primordial non-Gaussianity in a formalism of perturbation theory with a re-summation technique based on the Lagrangian picture (seealso [11, 12]), and the standard perturbation theory (SPT)combined with the phenomenological model of local bias,which leads to nontrivial renormalizations of the leading-order bias parameter [13]. A significant advantage of the iPTis that it is simpler and easier to use to calculate the powerspectrum than other resummation methods even in the pres-ence of halo bias and redshift-space distortions. The compu-tational cost is similar to that of the SPT.We focus not on the power spectrum but on the two-pointcorrelation function, because we do not su ff er from shot noisee ff ect in the correlation function. While two-loop correctionsin the iPT generally might have an impact on the correlationfunction on quasilinear scales [14], we use one-loop iPT forsimplicity in this paper.This paper is organized as follows. We first review thetheoretical predictions of the power spectrum and correlationfunction in Section II. We describe the details of N -body sim-ulations and a method to compute the correlation functionsof matter and halos from N -body simulations in Section III.After showing the results of the matter and halo correlationfunctions and its cross-correlation function in Section IV, wethen show the main results of this paper in Section V. Finally,Section VI is devoted to our conclusion. II. ANALYTIC MODELS
In this section, we briefly review two theoretical models:the iPT model with nonlocal bias and the SPT model with lo-cal bias, which are compared with N -body simulation results. A. Predictions of integrated Perturbation Theory
We use iPT [10] to investigate how the cross-correlation co-e ffi cient behaves on quasilinear scales for various halo massesand redshifts. It is convenient to write down the power spec-trum predictions of the iPT based on multipoint propagatorsrecently introduced in Bernardeau et al. [15]. Using the multi-point propagators Γ ( n ) , the one-loop power spectrum betweenobject X and Y based on the iPT can be written as (the fullderivation is given in [16]) P XY ( k ) = Π ( k ) " ˆ Γ (1) X ( k ) Γ (1) Y ( k ) P L ( k ) + k π Z ∞ d r r Z − d x ˆ Γ (2) X ( k , r , x ) ˆ Γ (2) Y ( k , r , x ) P L ( kr ) P L (cid:16) k √ + r − rx (cid:17) , (1)where indices X and Y are either matter ’m’ or halo ’h’ in this paper, P L ( k ) is the linear matter power spectrum, and the vertexfactor Π ( k ) is given by Π ( k ) = exp " − k π Z d p P L ( p ) . (2)The normalized multipoint propagators ˆ Γ (1) and ˆ Γ (2) are given byˆ Γ (1)m ( k ) = + k π Z ∞ d r Z − d x " r (1 − x ) + r − rx +
37 (1 − rx )(1 − x ) rx + r − rx P L ( kr ) , (3)ˆ Γ (1)h ( k ) = + c (1)h ( k ) + k π Z ∞ d r Z − d x ( r (1 − x ) + r − rx +
37 (1 − rx )(1 − x )1 + r − rx h rx + r c (1)h ( kr ) i) P L ( kr ) , (4)ˆ Γ (2)m ( k , r , x ) = xr + − rx + r − rx −
47 1 − x + r − rx , (5)ˆ Γ (2)h ( k , r , x ) = xr h + c (1)h (cid:16) k √ + r − rx (cid:17)i + − rx + r − rx h + c (1)h ( kr ) i −
47 1 − x + r − rx + c (2)h (cid:16) kr , k √ + r − rx (cid:17) , (6)where indices ’m’ and ’h’ denote the matter and halo, andwe assume that the second-order renormalized bias function c (2)h ( k , k ) depends only on the magnitudes of the wave vec-tors, k ≡ | k | and k ≡ | k | . From Equations (3)-(6), wecan easily understand that the matter result is recovered when c (1)h = c (2)h =
0. Here c (1)h and c (2)h are renormalized bias func-tions in Lagrangian space introduced by Matsubara [10] andobtained as c ( n )h ( k , . . . , k n ) = Z ν ν f MF ( ν ) M ˆ c ( n )h ( k , . . . , k n ; ν )d ln ν Z ν ν f MF ( ν ) M d ln ν , (7)for a mass range M ≤ M ≤ M (see Equations 64 and 108of Matsubara [17]), where ν = δ c /σ ( M ) is a function of mass M , and δ c is the critical overdensity for spherical collapse.In an Einstein-de Sitter cosmology, the critical overdensity is δ c ≈ . σ ( M ) is the root-mean-square linear density field smoothed with a top-hat filter ofradius R enclosing an average mass M = ρ π R / σ ( M ) = Z k d k π W ( kR ) P L ( k ) , (8)with W ( x ) = j ( x ) x = x (sin x − x cos x ) , (9)where ρ is the mean matter density of the universe and j ( x )is the first-order spherical Bessel function. f MF ( ν ) is the scaleddi ff erential mass function defined as [20] f MF ( ν ) = M ρ n ( M ) d M d ln ν , (10)where n ( M ) is the comoving number density of halos withmass M . The quantity f MF ( ν ) is frequently used in the litera-ture and there have been several analytic predictions [21–23]and fitting formulas [e.g., 20, 24–29]. In this paper, we usethe fitting formula for the mass function introduced by Bhat-tacharya et al. [29], which shows better agreement with oursimulations [30]. ˆ c L n is given as (see, Equations 92, 95, and 96in Matsubara [17])ˆ c ( n )h ( k , . . . , k n ; ν ) = b L n ( M ) W ( k R ) · · · W ( k n R ) + A n − ( M ) δ nc dd ln σ ( M ) [ W ( k R ) · · · W ( k n R )] , (11)with A ( M ) = , (12) A ( M ) = + δ c b L1 ( M ) , (13)where b L n is the Lagrangian bias function for the halo bias.The theoretical two-point correlation function can be ex-pressed in terms of the power spectrum as ξ XY ( r ) = Z k d k π sin ( kr ) kr P XY ( k ) . (14) B. Standard perturbation theory with local bias model
In the SPT formalism, we consider the local determinis-tic nonlinear biasing model. Following Fry and Gaztanaga[31], we restrict the consideration on large scales in Eulerianspace and assume that the halo density can be described bya smoothed function F ( δ m ) that depends only on the matterdensity. We can expand F in a Taylor series around δ m suchthat δ h = F ( δ m ) = ∞ X n = b E n n ! δ n m , (15)where δ m is the nonlinear matter density. We then combinethis expansion with SPT, which expands the matter densityperturbations into a series δ m = δ (1)m + δ (2)m + · · · , where δ (1)m is the linear density field and δ ( n )m is of order [ δ (1)m ] n . At thenext-to-leading order, we can obtain the auto- and cross-powerspectrum of halos as [13, 32] P hh ( k ) = b P NL ( k ) + b b A ( k ) + b B ( k ) + N , (16) P hm ( k ) = b P NL ( k ) + b A ( k ) , (17)where b and b are the renormalized bias parameters, N is therenormalized shot noise, and P NL ( k ) is the nonlinear matterpower spectrum. b and b should be determined empiricallyor treated as free parameters. In this paper, we will exam-ine both cases in Section V. The terms A ( k ) and B ( k ) can beobtained as A ( k ) = Z d q (2 π ) F ( q , k − q ) P L ( q ) P L ( | k − q | ) , (18) B ( k ) = Z d q (2 π ) P L ( q ) (cid:2) P L ( | k − q | ) − P L ( q ) (cid:3) , (19) where F is the second-order mode coupling kernel in SPT, F ( k , k ) = + k · k k k k k + k k ! + k · k k k ! . (20)Taking Fourier transforms, we then obtain correspondingcorrelation functions given by ξ hh ( r ) = b ξ NL ( r ) + b b A ( r ) + b B ( r ) , (21) ξ hm ( r ) = b ξ NL ( r ) + b A ( r ) , (22)where ξ NL is the nonlinear matter correlation function, and A ( r ) and B ( r ) are the Fourier transforms of A ( k ) and B ( k ).Note that B ( r ) = ξ ( r ) − σ δ D ( r ) where ξ L ( r ) is the linear mat-ter correlation function, σ = R d q P ( q ) / (2 π ) , and δ D ( r ) isthe Dirac delta function. III. N -BODY SIMULATIONSA. Simulation parameters To obtain accurate predictions of the cross-correlation co-e ffi cient, we resort to the use of high-resolution N -body sim-ulations of structure formation. To perform the N -body sim-ulations, we use a publicly available tree-particle mesh code, Gadget2 [33]. We adopt the standard Λ CDM model with thematter density Ω m = . Ω b = . Ω Λ = .
735 with equation of state pa-rameter w = −
1, the spectral index n s = . h − Mpc σ = .
80, and the Hubble parameter h = .
71. These cos-mological parameters are consistent with the Wilkinson Mi-crowave Anisotropy Probe 7-year results [34]. We performedtwo types of simulations, both with N p = particles incubic boxes. The first type has a side L box = h − Mpcwith softening length r s being 50 h − kpc, and the second typehas a side L box = h − Mpc with softening length r s being100 h − kpc. These two types are named as L1000 and L2000,respectively. The initial conditions are generated based on thesecond-order Lagrangian perturbation theory [35, 36] with theinitial linear power spectrum calculated by CAMB [37]. Theinitial redshift is set to z ini =
36 for L1000 and z ini =
31 forL2000. We perform N run =
30 and 10 realizations for L1000and L2000, respectively. We summarize the simulation pa-rameters in Table I. The L1000 simulations used in this paperare the same as L1000 used in Sato and Matsubara [30].We store outputs at z = .
0, 2.0, 1.0, 0.5, 0.3, and 0and identify halos for each output using a Friends-of-Friends(FOF) group finder with linking length of 0.2 times the meanseparation [38]. We select halos in which the number of par-ticles, N p , is equal to or larger than 20 which correspondsto the halos with masses 1 . × h − M ⊙ for L1000 and1 . × h − M ⊙ for L2000. Then we divide halos into fivemass bins to keep track of their di ff erent clustering properties.The average number and mass of halos among realizations forredshifts are listed in Table II. The halo catalogs of Bin 4 in TABLE I: Parameters in high- and low-resolution N -body simulations: the matter density Ω m , the dark energy density Ω Λ , the baryon density Ω b , the Hubble parameter h , the spectral index n s , the variance of the density perturbations at 8 h − Mpc σ , the box size L box , the number ofparticles N p , the initial redshift z ini , the softening length r s , and the number of realizations N run .Name Ω m Ω Λ Ω b h n s σ L box N p z ini r s N run L1000 (high resolution) 0.265 0.735 0.0448 0.71 0.963 0.80 1000 h − Mpc 1024
36 50 h − kpc 30L2000 (low resolution) 0.265 0.735 0.0448 0.71 0.963 0.80 2000 h − Mpc 1024
31 100 h − kpc 10TABLE II: Properties of halo catalogs of high- and low-resolution N -body simulations for each mass bin. We use the five halo catalogsabbreviated as “Bin 1”, . . . , “Bin 5”. ¯ N h and ¯ M h are the average halo numbers and average halo masses at various redshifts. L1000
Bin 1 Bin 2 Bin 31 . ≤ M h / (10 h − M ⊙ ) < .
11 4 . ≤ M h / (10 h − M ⊙ ) < .
32 1 . ≤ M h / (10 h − M ⊙ ) < . z ¯ N h ¯ M h [ h − M ⊙ ] z ¯ N h ¯ M h [ h − M ⊙ ] z ¯ N h ¯ M h [ h − M ⊙ ]3 . × × . × × . × × . × × . × × . × × . × × . × × . × × . × × . × × . × × . × × . × × . × × × × × × × × L2000
Bin 4 Bin 51 . ≤ M h / (10 h − M ⊙ ) < .
70 3 . ≤ M h / (10 h − M ⊙ ) < . z ¯ N h ¯ M h [ h − M ⊙ ] z ¯ N h ¯ M h [ h − M ⊙ ]3 . × × . × × . × × . × × . × × . × × . × × . × × . × × . × × × × × × L2000 is constructed so that the halo mass range is the same asthat of Bin 3 in L1000, as shown in Table II. Since the volumeof L2000 simulations is bigger than that of L1000 simulations,the number of halos with a certain mass are larger for L2000simulations.
B. Analysis: two-point correlation functions
To calculate the two-point correlation function of dark mat-ter from N -body simulations, we first randomly choose thenumber of particles N p , r = and 256 for L1000 andL2000. For dark matter halos, we use all halos in each bin.Then we directly count the N -body particle and / or halos tocalculate the two-point correlation function instead of usingthe fast Fourier transform method [30]. We choose r i to be thecenter of the i th bin, i.e., r i = ( r min i + r max i ) /
2, where r min i and r max i are the minimum and maximum distances of the i th bin. The shot noise corrections in the halo power spectrum aresubtle. If the dark matter halos are regarded as a Poisson pro-cess, we can easily subtract the shot noise e ff ect by using thenumber density of halos ¯ n h . However, Smith et al. [39] foundthat this standard correction method is not exactly correct forhalos, particularly for those of large mass. This is probablybecause in order to identify halos using the FOF algorithm, weautomatically impose that distances between halos are largerthan the sum of their radii, or they would have been linked asbigger halos. Thus, the shot noise e ff ect is scale dependentand it is di ffi cult to correctly subtract the e ff ect of shot noise.Therefore, we use the correlation function instead of using thepower spectrum, because the shot noise e ff ect in the correla-tion function is weaker than that in the power spectrum. FIG. 1:
Top panel : Correlation functions of matter, halo, and theircross-correlation function at redshift z =
0, multiplied by a factor of r . For the halo mass range, we consider 1 . ≤ M h / (10 h − M ⊙ ) < .
11 (Bin 1). The symbols are the results obtained from N -bodysimulations. The solid lines are the results of integrated perturba-tion theory (iPT) [10]. Bottom panel : Ratio of correlation functionsmeasured from N -body simulations to those from the iPT. IV. CORRELATION FUNCTIONS
Before presenting the results for the cross-correlation co-e ffi cient, we compare the N -body simulation results with theiPT for the correlation functions themselves.Figure 1 shows the results for the correlation functions ofmatter and halos, and their cross-correlation function at z = b E1 in this halorange is 0.904 (less than 1). The error bars describe the 1- σ error on the mean values obtained from 30 realizations. Theerror bars increase on large scales because of the finite sizeof the simulation box. The iPT predictions agree with N -bodysimulation results down to r ∼ h − Mpc within a few percentfor all correlations. In Section V, we will see that a range ofa few percent-level agreement in the cross-correlation coe ffi -cient is extended more than that in the correlation functions. V. CROSS-CORRELATION COEFFICIENT
In the framework of the local biasing model, the densityfield of galaxies and their halos should be a stochastic functionof the underlying dark matter density field [9]. The stochas-ticity is very weak on large scales, while it becomes more im-portant on small scales [40–43].One of the characteristic parameters of stochasticity is the cross-correlation coe ffi cient between the matter and halo den-sity fields, defined as r cc ( r ) = ξ hm ( r ) p ξ mm ( r ) ξ hh ( r ) , (23)where ξ mm ( r ), ξ hh ( r ), and ξ hm ( r ) are the matter and halo auto-correlation functions, and their cross-correlation function, re-spectively. The cross-correlation coe ffi cient is the measureof the statistical coherence of the two fields [44–49]. If anyscale-dependent, deterministic, linear-bias model is assumed,we have r cc =
1. Therefore, deviations of the cross-correlationcoe ffi cient from unity would arise due to both the nonlinearityand stochasticity of bias.Figure 2 shows the cross-correlation coe ffi cient betweenthe matter and halo density fields at z =
0, 0.3, 0.5, 1.0,2.0 and 3.0. The cross, square, and triangle symbols arethe N -body simulation results measured from 30 realizationsfor halo masses 1 . ≤ M h / (10 h − M ⊙ ) < .
11 (Bin 1),4 . ≤ M h / (10 h − M ⊙ ) < .
32 (Bin 2), and 1 . ≤ M h / (10 h − M ⊙ ) < .
70 (Bin 3), respectively. The error barsdescribe the 1- σ error on the mean value obtained from 30 re-alizations. We do not plot the results in which the sum of the1- σ error bars in a range of 5 ≤ r i / ( h − Mpc) ≤
100 is largerthan 0.12, i.e., P ≤ r i ≤ σ i > .
12. It should be noted that ha-los in each bin are more biased as redshift increases, becausewe impose the same halo mass ranges for each bin. The solidcurves show the iPT predictions. The iPT obtains good agree-ments with simulation results down to r ∼ h − Mpc withina range of error bars for all redshifts and halo mass ranges wehave considered. Particularly at z = .
0, the iPT well repro-duces the simulation result down to r ∼ h − Mpc. The devia-tions from unity in the cross-correlation coe ffi cient r cc on largescales are physical e ff ects. Similar e ff ects were also predictedeven in a simple model of local bias by Scherrer and Wein-berg [50]. The iPT prediction for the deviations has the sameorigin as theirs: the nonlinear dynamics on small scales non-trivially a ff ect the cross coe ffi cients on very large scales. Oursimulations are consistent with these theoretical predictions.Below, we will see fractional di ff erences between simulationresults and theoretical predictions in Figure 5, to discuss thepercentage error. The di ff erence between the iPT and simula-tion results on small scales probably comes from the fact thatthe iPT breaks down on small scales (see, Figure 1) [30, 51].One can see that the iPT prediction on small scales is almostflat, unlike the simulation results. This is probably because theasymptotic behaviors of the correlation functions based on theiPT are almost the same (see Figure 1), and at any rate the iPTshould not be applied on such small scales.We also plot a simple model derived from Equations (21)and (22) as dotted curves and it is expressed as [32] r cc ( r ) = − b b ! ξ L ( r ) , (24)by using the approximations A ( r ) ≪ ξ NL ( r ) and B ( r ) ≪ ξ NL ( r ).To empirically estimate b and b , we use general relationsbetween local bias parameters in Lagrangian space and Eule-rian space, which are derived in the spherical collapse model FIG. 2: The cross-correlation coe ffi cient between the matter and halo density fields at z =
0, 0.3, 0.5, 1.0, 2.0, and 3.0. For the halo massranges, we consider 1 . ≤ M h / (10 h − M ⊙ ) < .
11 (Bin 1), 4 . ≤ M h / (10 h − M ⊙ ) < .
32 (Bin 2), and 1 . ≤ M h / (10 h − M ⊙ ) < . σ error bars in a range of 5 ≤ r i / ( h − Mpc) ≤
100 is larger than 0.12, i.e., P ≤ r i ≤ σ i > .
12. The symbols are the results measured from N -body simulations. The solid lines are the results of integrated perturbationtheory (iPT) [10] while the dotted lines are the results of the simple model derived from standard perturbation theory with local bias model(Equation 24). To empirically estimate b and b , we use the relation in Equations (25) and (26) and then simply substitute b E1 and b E2 with b and b . FIG. 3: Same as Figure 2, but for the dotted lines, we fit b / b to the N -body simulation results using a chi-square fit. FIG. 4: The cross-correlation coe ffi cients with halo mass ranges 1 . ≤ M h / (10 h − M ⊙ ) < .
70 (Bin 4) and 3 . ≤ M h / (10 h − M ⊙ ) < .
09 (Bin 5), given at redshift z =
0, 0.3, 0.5, and 1.0. We do not show the results in which the sum of the 1- σ error bars in a range of5 ≤ r i / ( h − Mpc) ≤
100 is larger than 0.12. The triangle and circle symbols are the results of N -body simulations. The solid lines correspondto the results of integrated perturbation theory (iPT) [10] while the dotted lines correspond to results of standard perturbation theory with thefitted bias model (Equation 24). as [10] b E1 = b L1 + , (25) b E2 = b L1 + b L2 , (26)where b E1 and b E2 are Eulerian bias parameters. Note that boththe Eulerian bias parameters b E n and the Lagrangian bias pa-rameters b L n are local and independent of scales. In this phe-nomenological model, we simply substitute b E1 and b E2 with b and b . To calculate b L1 and b L2 , we use [12] b L n = ( − n δ nc Z M M ν n d n f MF ( ν )d ν n d ln σ ( M )d M d MM Z M M f MF ( ν ) d ln σ ( M )d M d MM . (27)for halos in a mass range M ≤ M ≤ M . The simple model(Equation 24) with the above estimates of bias parametersshows better agreement with simulations for higher redshifts(i.e., more biased halos). We find that the cross-correlation FIG. 5: Fractional di ff erences between N -body results and theoretical predictions are shown in percents, at redshifts z =
0, 0.3, 0.5, 1.0, 2.0,and 3.0, from bottom to top . Left panels: iPT,
Middle panels:
SPT, and
Right panels:
SPT with fitting. The red cross, blue box, and greentriangles are the results of Bin 1, 2, and 3. coe ffi cients of halos with b L1 & b and b are treated as free parameters,we fit b / b to the simulation results using a chi-square fit.The result is shown as dotted lines in Figure 3. Other linesand symbols are the same as in Figure 2. Fittings are done ina range of 5 ≤ r / ( h − Mpc) ≤
70. In the fitting case, an im-provement from the above empirical method is little for cases of high bias, but is important for cases of low bias. The simplemodel with fitted bias replicates the simulation results over allscales at 1 ≤ z ≤
3. We can see that the cross-correlationcoe ffi cients estimated from N -body simulations have compli-cated behaviors in quasilinear regimes at low redshifts, whichcannot be described in the simple model. We will describepercentage error later in Figure 5.Figure 4 shows the results for the cross-correlation coe ffi cient of large halos with massranges 1 . ≤ M h / (10 h − M ⊙ ) < .
70 (Bin 4) and0
FIG. 6: Same as Figure 5, but the results of Bin 4 (green triangle) and 5 (magenta circle) at z =
0, 0.3, 0.5 and 1.0. . ≤ M h / (10 h − M ⊙ ) < .
09 (Bin 5) at redshift z = b / b are not, in general, the same as those obtained fromother statistics, such as the power spectrum and bispectrum,because b and b are renormalized. To clarify how well theoretical models predict the N -body results, we plot fractional di ff erences between N -bodysimulation results and theoretical predictions, [ r cc , sim ( r ) − r cc , theo ( r )] / r cc , theo ( r ), as shown in Figures 5 and 6. These fig-ures show that the iPT agrees with simulation results down to r ∼
15 (10) h − Mpc within 0.5 (1.0) % for all redshifts andhalo masses we considered. It should be noted that the iPTdoes not have any fitting parameter. The SPT with empiri-cally determined bias reproduces N -body simulation resultsdown to r ∼ h − Mpc within a percent-level for all redshiftsexcept for z = b and b are di ff erent from b E and b E , which canbe determined by other methods, e.g., the power spectrum andbispectrum. VI. CONCLUSION
In this paper, we have used 40 large cosmological N -body simulations of the standard Λ CDM cosmology to in-vestigate the cross-correlation coe ffi cient between the haloand matter density fields over a wide redshift range. Thecross-correlation coe ffi cient is crucial to extract informationof the matter density field by combining galaxy clustering andgalaxy-galaxy lensing measurements. Since the first attemptto detect galaxy-galaxy lensing [52], its ability to constraincosmological parameters has been shown [8].We compared the simulation results with theoretical predic-tions of the iPT and simple models of bias with SPT. The iPTpredicts the simulation results down to r ∼
15 (10) h − Mpcwithin 0.5 (1.0) % for all redshifts and halo masses we con-sidered. To improve the prediction, the two-loop correction tothe iPT might be important. In the SPT with local bias model,bias parameters are renormalized and therefore they are de-termined empirically or treated as free parameters. The SPTwith empirically determined biases with the spherical collapsemodel shows better agreement with simulations for more bi-ased halos on small scales, although this model does not re-produce the complicated behaviors of the simulation resultson quasilinear scales at low redshifts. The SPT with biases de-termined by fitting improves the predictions but the situationis almost the same at low redshift. Thus, the iPT accuratelypredicts the cross-correlation coe ffi cient as long as quasilinear scales are considered.Let us finally comment on convolution Lagrangian pertur-bation theory (CLPT), which was recently proposed by Carl-son et al. [51]. The CLPT applies additional resummations ontop of the simple LRT (restricted iPT with local Lagrangianbias), and its prediction significantly improves the simple LRTfor the correlation function in real and redshift spaces on smallscales. Therefore, it might be possible that the CLPT gives abetter prediction for the cross-correlation coe ffi cient betweenmass and halos and agrees with simulation results on smallscales. Although it is important to examine how well theCLPT predicts these results, we leave it for future work.In this paper we focused on fundamental features of biasstochasticity by the methods of numerical simulations and the-oretical models. We believe the results of this paper could bea crucial first step to understand the galaxy biasing for futureprecision cosmology. Acknowledgments
We thank Uroˇs Seljak for useful comments. M.S. is sup-ported by a Grant-in-Aid for the Japan Society for Promotionof Science (JSPS) fellows. T.M. acknowledges support fromthe Ministry of Education, Culture, Sports, Science, and Tech-nology (MEXT), Grant-in-Aid for Scientific Research (C),No. 24540267, 2012. This work is supported in part by aGrant-in-Aid for Nagoya University Global COE Program,“Quest for Fundamental Principles in the Universe: from Par-ticles to the Solar System and the Cosmos”, from the MEXTof Japan. Numerical computations were in part carried out onCOSMOS provided by Kobayashi-Maskawa Institute for theOrigin of Particles and the Universe, Nagoya University. [1] M. R. Blanton, D. Eisenstein, D. W. Hogg, and I. Zehavi, ApJ , 977 (2006).[2] W. J. Percival, R. C. Nichol, D. J. Eisenstein, J. A. Frieman,M. Fukugita, J. Loveday, A. C. Pope, D. P. Schneider, A. S.Szalay, M. Tegmark, et al., ApJ , 645 (2007).[3] A. G. S´anchez and S. Cole, MNRAS , 830 (2008).[4] C. Blake, S. Brough, M. Colless, C. Contreras, W. Couch,S. Croom, T. Davis, M. J. Drinkwater, K. Forster, D. Gilbank,et al., MNRAS , 2876 (2011).[5] D. Schlegel, F. Abdalla, T. Abraham, C. Ahn, C. Allende Prieto,J. Annis, E. Aubourg, M. Azzaro, S. B. C. Baltay, C. Baugh,et al., arXiv:1106.1706 (2011).[6] R. Laureijs, J. Amiaux, S. Arduini, J. . Augu`eres, J. Brinch-mann, R. Cole, M. Cropper, C. Dabin, L. Duvet, A. Ealet, et al.,arXiv:1110.3193 (2011).[7] R. Ellis, M. Takada, H. Aihara, N. Arimoto, K. Bundy,M. Chiba, J. Cohen, O. Dore, J. E. Greene, J. Gunn, et al.,arXiv:1206.0737 (2012).[8] R. Mandelbaum, A. Slosar, T. Baldauf, U. Seljak, C. M. Hi-rata, R. Nakajima, R. Reyes, and R. E. Smith, arXiv:1207.1120(2012).[9] A. Dekel and O. Lahav, ApJ , 24 (1999).[10] T. Matsubara, Phys.Rev.D , 083518 (2011). [11] T. Matsubara, Phys.Rev.D , 063530 (2008).[12] T. Matsubara, Phys.Rev.D , 083519 (2008).[13] P. McDonald, Phys.Rev.D , 103512 (2006).[14] T. Okamura, A. Taruya, and T. Matsubara, JCAP , 012 (2011).[15] F. Bernardeau, M. Crocce, and R. Scoccimarro, Phys.Rev.D ,103521 (2008).[16] T. Matsubara, arXiv:1304.4226 (2013).[17] T. Matsubara, Phys.Rev.D , 063518 (2012).[18] T. T. Nakamura and Y. Suto, Progress of Theoretical Physics , 49 (1997).[19] J. P. Henry, ApJ , 565 (2000).[20] A. Jenkins, C. S. Frenk, S. D. M. White, J. M. Colberg, S. Cole,A. E. Evrard, H. M. P. Couchman, and N. Yoshida, MNRAS , 372 (2001).[21] W. H. Press and P. Schechter, ApJ , 425 (1974).[22] J. R. Bond, S. Cole, G. Efstathiou, and N. Kaiser, ApJ , 440(1991).[23] R. K. Sheth, H. J. Mo, and G. Tormen, MNRAS , 1 (2001).[24] R. K. Sheth and G. Tormen, MNRAS , 119 (1999).[25] M. S. Warren, K. Abazajian, D. E. Holz, and L. Teodoro, ApJ , 881 (2006).[26] D. S. Reed, R. Bower, C. S. Frenk, A. Jenkins, and T. Theuns,MNRAS , 2 (2007). [27] M. Crocce, P. Fosalba, F. J. Castander, and E. Gazta˜naga, MN-RAS , 1353 (2010).[28] M. Manera, R. K. Sheth, and R. Scoccimarro, MNRAS ,589 (2010).[29] S. Bhattacharya, K. Heitmann, M. White, Z. Luki´c, C. Wagner,and S. Habib, ApJ , 122 (2011).[30] M. Sato and T. Matsubara, Phys.Rev.D , 043501 (2011).[31] J. N. Fry and E. Gaztanaga, ApJ , 447 (1993).[32] T. Baldauf, R. E. Smith, U. Seljak, and R. Mandelbaum,Phys.Rev.D , 063531 (2010).[33] V. Springel, MNRAS , 1105 (2005).[34] E. Komatsu, K. M. Smith, J. Dunkley, C. L. Bennett, B. Gold,G. Hinshaw, N. Jarosik, D. Larson, M. R. Nolta, L. Page, et al.,ApJS , 18 (2011).[35] M. Crocce, S. Pueblas, and R. Scoccimarro, MNRAS , 369(2006).[36] P. Valageas and T. Nishimichi, A&A , A87 (2011).[37] A. Lewis, A. Challinor, and A. Lasenby, ApJ , 473 (2000).[38] M. Davis, G. Efstathiou, C. S. Frenk, and S. D. M. White, ApJ , 371 (1985). [39] R. E. Smith, R. Scoccimarro, and R. K. Sheth, Phys.Rev.D ,063512 (2007).[40] T. Matsubara, ApJ , 543 (1999).[41] A. Taruya and Y. Suto, ApJ , 559 (2000).[42] K. Yoshikawa, A. Taruya, Y. P. Jing, and Y. Suto, ApJ , 520(2001).[43] Y.-C. Cai, G. Bernstein, and R. K. Sheth, MNRAS , 995(2011).[44] U.-L. Pen, ApJ , 601 (1998).[45] M. Tegmark and B. C. Bromley, ApJ , L69 (1999).[46] M. Tegmark and P. J. E. Peebles, ApJ , L79 (1998).[47] U. Seljak and M. S. Warren, MNRAS , 129 (2004).[48] S. Bonoli and U. L. Pen, MNRAS , 1610 (2009).[49] M. Cacciato, O. Lahav, F. C. van den Bosch, H. Hoekstra, andA. Dekel, MNRAS , 566 (2012).[50] R. J. Scherrer and D. H. Weinberg, ApJ , 607 (1998).[51] J. Carlson, B. Reid, and M. White, MNRAS , 1674 (2013).[52] J. A. Tyson, F. Valdes, J. F. Jarvis, and A. P. Mills, Jr., ApJ281