Volume 542, June 2012
|Number of page(s)||31|
|Section||Cosmology (including clusters of galaxies)|
|Published online||24 May 2012|
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. Dark-matter 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 δc, 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 δc 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 top-hat filter of width , computed from linear theory, (A.5)where W(x) = (3/x3)[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 Mlow = 103 h-1 M⊙ and Mhigh = 1016 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 = rvir/rs is the “concentration parameter”, for which we assume the following expression, (A.10)We take c0 = 11 and β = 0.13, and M ⋆ is defined such that ν(z = 0) = 1, i.e. δc(0) = σ(M ⋆ ). The virial radius rvir is given by the following relation (A.11)with Δvir(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 scale-dependent halo bias (A.13)where ξm is the matter correlation function. The large-scale halo bias bh(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 one-halo term, to express the galaxy correlation inside a halo, and the two-halo term, to account for halo-to-halo correlation, (A.15)the one-halo term depends on the number of galaxy pairs ⟨ N(N − 1) ⟩ per halo. This is comprised of the central-satellite contribution ⟨ NcNs ⟩ and the satellite-satellite term ⟨ Ns(Ns − 1) ⟩ . Assuming a Poisson distribution, we write (A.16)The one-halo term correlation function for central-satellite pairs is then (A.17)The lower integration limit Mvir(r) is the virial mass contained in a halo of radius r, computed with Eq. (A.11). This accounts for the fact that less-massive haloes are too small to contribute to the correlation at separation r.
The one-halo satellite-satellite contribution ξss involves the halo profile auto-convolution and is therefore easier to compute in Fourier space. The corresponding power spectrum is written as (A.18)where uh(k|M) is the Fourier transform of the dark-matter halo profile ρh(r|M). The correlation function ξss is then obtained via a Fourier transform. The one-halo correlation function is the sum of the two contributions, (A.19)The two-halo term is derived from the dark-matter power spectrum and the halo two-point correlation function: (A.20)where (A.21)The upper integration limit Mlim(r) takes into account the halo exclusion (Zheng 2004), i.e. the fact that haloes are
non-overlapping. We follow Tinker et al. (2005) to compute Mlim(r) by matching with the following expression, (A.22)where P(r,M1,M2) is the probability that two ellipsoidal haloes of mass M1 and M2, respectively, do not overlap. Defining x = r/[rvir(M1) + rvir(M2)] 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 non-overlapping haloes to be (A.23)We Fourier-transform Eq. (A.20) for a range of tabulated values of r, to compute the two-halo term ξ2 of the galaxy autocorrelation function. Finally, we renormalise it to the total number of galaxy pairs: (A.24)The angular two-point 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: Best-fitting HOD parameters and deduced quantities
Description of all galaxy samples and best-fitting HOD parameters. Halo masses are given in h-1 M⊙ and galaxy number densities in h-3 Mpc3.
Description of red galaxy samples and best-fitting HOD parameters. Halo masses are given in h-1M⊙ and galaxy number densities in h-3 Mpc3.
Description of blue galaxy samples. Galaxy number densities are given in h-3 Mpc3.
Appendix C: Two-point correlation function measurements
Two-point correlation function measurements in the range 0.2 < z < 0.4.
Two-point correlation function measurements in the range 0.4 < z < 0.6.
Two-point correlation function measurements in the range 0.6 < z < 0.8.
Two-point correlation function measurements in the range 0.8 < z < 1.0.
Two-point correlation function measurements in the range 1.0 < z < 1.2.
1D (diagonal) and 2D likelihood distributions of best-fitting HOD parameters for the full sample, in the range 0.4 < z < 0.6, and Mg − 5log h < −19.8.
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