Halo Occupation Distribution

correlation functions probe also higher-order moments of the HOD in this formalism and they are defined accordingly. Note that the probability is normalized such that P∞

N=0P(N|m) = 1. Note that by using the factorial moments we neglect self-correlations of galaxy pairs, triplets, etc., which lead to shot noise terms. We need to add these shot noise terms because in contrast to the continuous dark matter field the observed galaxy distribution is a discrete field.

The only missing component for calculating galaxy spectra explicitly is a model for the HOD. The next section provides a comprehensive description of the HOD including its parametrization. Using these results we are able to model the galaxy auto- and galaxy-dark matter cross-spectra at arbitrary order. The results up to the fourth order which are needed for the rest of this thesis will be presented in subsequent sections.

i.e., hN(m)i ' 1. Considering a halo threshold mass of m = 10¹¹h⁻¹M the plateau extends to m = 10¹³h⁻¹M. For m > 10¹³h⁻¹M the halo is populated by satellite galaxies which are well described by a power law in halo mass. Note that the stated halo masses shift accordingly with increasing of the threshold mass.

Results of different simulations for higher-order moments of the HOD find that the distribution P(N|m) is sub-Poissonian (that is, the distribution is narrower than a Poisson distribution, or more precisely the corresponding variance is smaller than the variance of a Poisson distribution) around the threshold mass and Poissonian⁵ for large host halo masses. Several simple functions have been considered for the shape of the HOD distribution, and we present now the two most important ones.

The first results to model higher-order moments of the HOD were done using halo model fits to results of the APM (Automatic Plate Measuring Facility) galaxy survey (Scoccimarro et al. 2001). They assumed that the HOD is described by a binomial distribution, in which case higher orders are completely specified by the first and second moments of this distribution. In addition, they relate the second-order moment to the first-order moment by introducing the function α(m) which fulfills

hN(N −1)(m)i ≡α²(m)hN(m)i², (4.10) and is given by

α(m) =

(1 for m≥10¹³h⁻¹M, log₁₀

pm/m₁₁

for m <10¹³h⁻¹M. (4.11) In this model the threshold mass is m₁₁ = 10¹¹h⁻¹M. The function α(m) describes the departure from a Poisson distribution of the HOD. We see that for large masses the galaxies follow a Poisson distribution, but for small masses there are deviations from the Poisson statistics.

The second approach developed by Kravtsov et al. (2004) separates the number of galaxies in halos into two different contributions: central galaxies hosting the dark matter halo and a number of satellite galaxies that subsequently populate the halo.

The motivation for this decomposition comes partly from results of simulations and studies of central elliptical galaxies in groups and clusters which are often considered as a separate population from the rest of the galaxies in the observed group or cluster⁶. This separation turns out to naturally reproduce the observed sub-Poissonian behavior for small masses where the correlations of central galaxies dominate, as well as the change to the Poissonian behavior for large masses where satellite correlations dominate.

5For a Poisson distributed random variableX the ensemble average ofX is equal to the variance of X, i.e.,hXi=hX²i − hXi².

6The galaxy sitting in the center of the cluster is in the majority of cases the most luminous and massive one in the cluster and therefore called thebrightest cluster galaxy (BCG). These galaxies are generally old ellipticals and except for those undergoing major mergers, they lie at the bottom of the cluster potential well and close to the X-ray emission peak.

Furthermore, Kravtsov et al. assume that each halo hosts one central galaxy only if it has a mass above the minimal mass m_min. Below this minimal mass there will not be enough cold gas to form galaxies as mentioned before. Therefore, the number of central galaxies N_cen(m) can be approximated by a step function:

N_cen(m) = Θ(m−m_min), (4.12)

where Θ(m) is the Heaviside step function. In practice flux-limited galaxy samples are defined by a lower luminosity threshold L_min rather than a minimal host halo mass.

Therefore, we need to find a conversion of L_min tom_min. The stochastic nature of galaxy formation induces scatter in the central galaxy luminosity at a fixed halo massm which is well modeled by a log-normal distribution. This scatter in the luminosity translates into a scatter in the host halo mass. We can model the scatter by convolving the step function with a log-normal distribution of variance σ_ln² _m and zero mean. The result is⁷ (e.g., Zheng et al. 2007)

hN_cen(m)i= 1 2erfc

ln(m_min/m)

√2σ_lnm

. (4.13)

This more realistic modeling comes at the expense of an additional parameter, i.e., the dispersionσlnm. Using this approach, on average half of the halos in the sample with a mass of m =m_min host a central galaxy.

After the placement of the central galaxy the halo is populated by satellite galaxies.

The mean number of satellite galaxies follows approximately a Poisson distribution, where the first moment is well described by

hN_sat(m)i= m

m₁ β

Θ(m−m_min), (4.14)

where we use a simple power law with the amplitude and slope as free parameters.

Here m₁ is linearly related to the minimal mass, i.e., m₁ =A_sm_min with the parameter A_s >1. In summary, the HOD is characterized by four parameters: the minimal mass mmin above which halos are populated with galaxies, the scatter of the mean number of central galaxies σ_lnm, and the two parametersβ andA_s that specify the power law of satellite galaxies. Note that the minimal mass is not a free parameter but is fixed by matching the mean number density in Eq. (4.2)to that of a given survey sample defined by a minimal luminosity. Recent simulations indicate (Zheng et al. 2005) that at low masses the mean number of satellite galaxies drops below the power-law extrapolation from high masses. This behavior is well modeled by introducing an additional cut-off parameter m₀ that can differ from the minimal mass of central galaxies. Then the first moment of the satellite galaxies in Eq. (4.14) is replaced by

hN_sat(m)i=

m−m₀ m⁰₁

β

Θ(m−m₀), (4.15)

7Sometimes the dispersionσ_lnm is defined including a factor√ 2.

with the two new parametersm⁰₁ and m₀. As this results only in a minor change in the galaxy power spectrum we will adopt the four-parameter model.

Combining the contributions from the central and satellite galaxies we get the total number of galaxies, i.e., N =N_cen+N_sat. Using the different statistical properties of centrals and satellites, we are able to relate higher-order factorial moments of the total number of galaxies to first-order moments of central and satellite galaxies. We show explicitly the results of the first four orders of the HOD:

hN(N −1)i=hN_sat(N_sat−1)i+hN_cen(N_cen−1)i+ 2hN_cenN_sati

=hN_sati²+ 2hN_cenihN_sati, (4.16) hN(N −1)(N −2)i=hN_sati³+ 3hN_cenihN_sati², (4.17) hN(N −1)(N −2)(N −3)i=hN_sati⁴+ 4hN_cenihN_sati³. (4.18) The distribution of central galaxies can be described by a nearest-integer or Bernoulli distribution as halos contain either one central galaxy or none. Therefore, all factorial moments involving only central galaxies vanish. The factorial moments of satellites are easily calculated remembering that they follow a Poisson distribution, e.g., hN_sat(N_sat− 1)i=hNsati². Furthermore, we need to deal with terms involving cross-correlations of central and satellite species. These are easily determined, since by definition N_cen = 0 implies that also N_sat = 0. Thus, the cross-correlations split into the averages of the individual components, e.g., hNcenNsati=hNcenihNsati.

Using the approach of separating contributions from central and satellite galaxies is much simpler than assuming a binomial distribution because we need only the first moment of each species to calculate higher-order moments. The Poissonian nature of the satellite HOD which is the key ingredient of the model was tested in Kravtsov et al.

(2004) up to the third-order moment, and observational results verify this behavior, e.g., Yang et al. (2005).

We still need to specify our input parameters for the HOD model which are provided by results from simulations and observations. Hydrodynamic N-body simulations and pure dark matter N-body simulations resolving subhalos find that the satellite HOD is approximately proportional to the mass of the host halo, i.e., β ≈1. Note that in this casem=m₁ is the characteristic mass where a halo hosts on average one satellite galaxy.

All parameters for the fiducial model of the HOD we use in our analysis are summarized in Table A.1 (Model A). The minimal mass of m_min = 10¹²h⁻¹M corresponds to a mean number density of galaxies ofn¯_g = 4.96×10⁻³h³Mpc⁻³. Note that the parameters describing the HOD (β,Asandσlnm) are cosmology- and redshift-independent. However, one expects a dependence on cosmological parameters that has not yet been quantified and is possibly small. One would need a large ensemble of hydrodynamical simulations with varying cosmological models to test for this case which has not been done up to now. The form of the first moment of the HOD using the fiducial model is shown in Fig. 4.1. One clearly sees the smoothed-out step function from the contribution of central galaxies and the power-law behavior of satellite galaxies. Also shown is the dimensionless form of the halo mass function for the two redshifts z = 0 and z = 1

10^-4 10^-3 10^-2 10^-1 10⁰ 10¹ 10²

10¹⁰ 10¹¹ 10¹² 10¹³ 10¹⁴ 10¹⁵ 10¹⁶

m [h^-1 M_⊙] (m_*/ρ- ) m n(m,z)

z=0 z=1

<N(m)>

Figure 4.1:First moment of the HOD, i.e.,hN(m)iusing the parameters of the fiducial model (see Model A in Table A.1) as a function of halo mass showing the individual contributions of central and satellite galaxies (see Eqs. 4.13 and 4.14, respectively). One can clearly distinguish the contributions from central galaxies around the threshold massm_min= 10¹²h⁻¹M (we also show a pure step function in magenta) and the power-law behavior of the satellite galaxies for large masses. Also shown is the dimensionless form of the halo mass function for the redshiftsz= 0 and z= 1 which bracket the range essentially probed by weak gravitational lensing observations.

which correspond approximately to the redshift range used for current cosmic shear experiments. We see that the number density of halos drops off significantly for masses around m = 10¹⁴h⁻¹M and m = 10¹⁵h⁻¹M for z = 1 and z = 0, respectively.

Shifting the threshold mass for the central galaxy to higher masses will move the HOD to higher masses. As the halo mass function has an exponential cutoff for high-mass halos, the satellite galaxies contribution will be suppressed.

Dependence of the HOD on Galaxy Color

Results from observations and simulations indicate that the HOD is strongly dependent on galaxy color. Hence, different samples that are divided by color, e.g., in blue and red galaxies, should yield a different form of the HOD. Fits to results from semi-analytic models of galaxy formation (see for example Kauffmann et al. 1999) yield the following

power laws for red and blue galaxies, which are parametrized in Sheth & Diaferio (2001):

hNblue(m)i= 0.7 m

m_blue α_b

, hNred(m)i= m

m_red αr

. (4.19)

For blue galaxies the power-law index is given by α_b =

(0 for mmin ≤m≤mblue,

0.8 for m > mblue, (4.20)

where we introduced the threshold mass m_min = 10¹¹h⁻¹M and the cutoff for blue galaxies mblue = 4×10¹²h⁻¹M. For red galaxies the power-law index is

α_r= 0.9, form≥m_min, (4.21)

wherem_red= 2.5×10¹²h⁻¹M. The total number of galaxies is then the sum of blue and red galaxies, i.e.,

hN(m)i=hN_blue(m)i+hN_red(m)i. (4.22) Note that these relations follow the HOD approach of Scoccimarro et al. outlined in the previous section and are not applicable for the Kravtsov et al. formalism which distinguishes between the contributions from central and satellite galaxies. For high halo masses the red galaxy HOD is steeper than the one for blue galaxies.

The parametrization of the HOD is not unique. A recent description of the HOD of early- and late-type galaxies can be found in Zheng et al. (2005).