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. 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 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 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 M_low = 10³ h^-1 M_⊙ and M_high = 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 = r_vir/r_s 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. 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 scale-dependent halo bias (A.13)where ξ_m is the matter correlation function. The large-scale 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 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  ⟨ N_cN_s ⟩  and the satellite-satellite term  ⟨ N_s(N_s − 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 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 less-massive haloes are too small to contribute to the correlation at separation r.

The one-halo satellite-satellite contribution ξ_ss 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 u_h(k|M) is the Fourier transform of the dark-matter halo profile ρ_h(r|M). The correlation function ξ_ss 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 M_lim(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 M_lim(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/[r_vir(M₁) + r_vir(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

Appendix C: Two-point correlation function measurements

	Fig. C.1 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.
Open with DEXTER

© ESO, 2012