Issue 
A&A
Volume 542, June 2012



Article Number  A5  
Number of page(s)  31  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201117625  
Published online  24 May 2012 
Online material
Appendix A: The halo model
The halo occupation distribution (HOD) model (Berlind & Weinberg 2002; Kravtsov et al. 2004; Zheng et al. 2005) allows one to derive physical information about galaxy populations and the dark matter haloes which host them. The HOD prescription is based on the halo model, which describes how dark matter is distributed in space. In this framework, all of the matter is assumed to reside in virialised haloes. The HOD parametrisation specifies how many galaxies populate haloes, on average, as function of halo mass. Accordingly, the number of galaxies per halo N only depends on the mass M of the halo.
A.1. Darkmatter halo model
The three ingredients to the halo model of dark matter are the halo mass function, the halo profile and the halo bias. The dark matter halo abundance can be inferred using the Press & Schechter (1974) approach where dark matter collapses into overdense regions above the critical density δ_{c}, linearly evolved to z = 0. The mass function, which is the halo number density per unit mass, can be parametrized as (A.1)where is the mean density of matter at the present day. The new mass variable ν writes (A.2)and characterizes the peak heights of the density field as function of mass and redshift. The linear critical density δ_{c} depends on the adopted cosmology and redshift; we use the fitting formula from Weinberg & Kamionkowski (2003); see also Kitayama & Suto (1996), D(z) is the linear growth factor at redshift z, and σ(M) is the rms of density fluctuations in a tophat filter of width , computed from linear theory, (A.5)where W(x) = (3/x^{3})[sinx − xcosx] . For the mass function f(ν), we choose the parameterisation by Sheth & Tormen (1999), calibrated on simulations: (A.6)where the normalisation A is fixed by imposing: (A.7)we adopt the values p = 0.3 and . If not indicated otherwise, all integrals over the mass function are performed between M_{low} = 10^{3} h^{1} M_{⊙} and M_{high} = 10^{16} h^{1} M_{⊙}.
We describe the halo density profile by the following form Navarro et al. (1997), (A.8)The total halo mass is then written as (see Takada & Jain 2003): (A.9)where c = r_{vir}/r_{s} is the “concentration parameter”, for which we assume the following expression, (A.10)We take c_{0} = 11 and β = 0.13, and M_{ ⋆ } is defined such that ν(z = 0) = 1, i.e. δ_{c}(0) = σ(M_{ ⋆ }). The virial radius r_{vir} is given by the following relation (A.11)with Δ_{vir}(z) being the critical overdensity for virialisation at redshift z (Kitayama & Suto 1996; Nakamura & Suto 1997; Henry 2000). We take the fitting formula from Weinberg & Kamionkowski (2003)(A.12)Following Tinker et al. (2005), we use the scaledependent halo bias (A.13)where ξ_{m} is the matter correlation function. The largescale halo bias b_{h}(M,z) is given by Sheth et al. (2001) as (A.14)As in Tinker et al. (2005), we adopt the revised parameters , b = 0.35 and c = 0.8.
A.2. The galaxy correlation function
We write the correlation function as a sum of two components: The onehalo term, to express the galaxy correlation inside a halo, and the twohalo term, to account for halotohalo correlation, (A.15)the onehalo term depends on the number of galaxy pairs ⟨ N(N − 1) ⟩ per halo. This is comprised of the centralsatellite contribution ⟨ N_{c}N_{s} ⟩ and the satellitesatellite term ⟨ N_{s}(N_{s} − 1) ⟩ . Assuming a Poisson distribution, we write (A.16)The onehalo term correlation function for centralsatellite pairs is then (A.17)The lower integration limit M_{vir}(r) is the virial mass contained in a halo of radius r, computed with Eq. (A.11). This accounts for the fact that lessmassive haloes are too small to contribute to the correlation at separation r.
The onehalo satellitesatellite contribution ξ_{ss} involves the halo profile autoconvolution and is therefore easier to compute in Fourier space. The corresponding power spectrum is written as (A.18)where u_{h}(kM) is the Fourier transform of the darkmatter halo profile ρ_{h}(rM). The correlation function ξ_{ss} is then obtained via a Fourier transform. The onehalo correlation function is the sum of the two contributions, (A.19)The twohalo term is derived from the darkmatter power spectrum and the halo twopoint correlation function: (A.20)where (A.21)The upper integration limit M_{lim}(r) takes into account the halo exclusion (Zheng 2004), i.e. the fact that haloes are
nonoverlapping. We follow Tinker et al. (2005) to compute M_{lim}(r) by matching with the following expression, (A.22)where P(r,M_{1},M_{2}) is the probability that two ellipsoidal haloes of mass M_{1} and M_{2}, respectively, do not overlap. Defining x = r/[r_{vir}(M_{1}) + r_{vir}(M_{2})] as the ratio of the halo separation and the sum of the virial radii, and y = (x − 0.8)/0.29, Tinker et al. (2005) found the probability of nonoverlapping haloes to be (A.23)We Fouriertransform Eq. (A.20) for a range of tabulated values of r, to compute the twohalo term ξ_{2} of the galaxy autocorrelation function. Finally, we renormalise it to the total number of galaxy pairs: (A.24)The angular twopoint correlation function w(θ) is computed from the observed photometric redshift distribution and ξ(r) using Limber’s equation (Limber 1954): (A.25)with (A.26)and x(z), the radial comoving coordinate.
Appendix B: Bestfitting HOD parameters and deduced quantities
Description of all galaxy samples and bestfitting HOD parameters. Halo masses are given in h^{1} M_{⊙} and galaxy number densities in h^{3} Mpc^{3}.
Description of red galaxy samples and bestfitting HOD parameters. Halo masses are given in h^{1}M_{⊙} and galaxy number densities in h^{3} Mpc^{3}.
Description of blue galaxy samples. Galaxy number densities are given in h^{3} Mpc^{3}.
Appendix C: Twopoint correlation function measurements
Twopoint correlation function measurements in the range 0.2 < z < 0.4.
Twopoint correlation function measurements in the range 0.4 < z < 0.6.
Twopoint correlation function measurements in the range 0.6 < z < 0.8.
Twopoint correlation function measurements in the range 0.8 < z < 1.0.
Twopoint correlation function measurements in the range 1.0 < z < 1.2.
Fig. C.1
1D (diagonal) and 2D likelihood distributions of bestfitting HOD parameters for the full sample, in the range 0.4 < z < 0.6, and M_{g} − 5log h < −19.8. 

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Bestfitting parameters (from Eq. (25)) of M_{min} and M_{1} as function of luminosity threshold corrected for passive redshift evolution to approximate stellar mass selected samples. Results are given for all and red samples, as function of redshift bins. Halo masses are given in h^{1} M_{⊙}.
© ESO, 2012
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