4.5 Halo occupation function approach for galaxy biasing

Since the clustering of dark matter halos is well understood now, one can describe the galaxy biasing if the halo model is combined with the relation between the halos and luminous objects. This is another approach to galaxy biasing, halo occupation function (HOF), which has become very popular recently. Indeed the basic idea behind HOF has a long history, but the model predictions have been significantly improved with the recent accurate models for the mass function, the biasing and the density profile of dark matter halos. We refer the readers to an extensive review on the HOF by Cooray and Sheth [13]. Here we briefly outline this approach.

We adopt a simple parametric form for the average number of a given galaxy population as a function of the hosting halo mass:

{ α (M ∕M1 ) for M > Mmin, Ng (M ) = 0 for M < Mmin. (125 )
The above statistical and empirical relation is the essential ingredient in the current modeling characterized by the minimum mass Mmin of halos which host the population of galaxies, a normalization parameter which can be interpreted as the critical mass M 1 above which halos typically host more than one galaxy (note that M1 may exceed Mmin since the above relation represents the statistical expected value of number of galaxies), and the power-law index α of the mass dependence of the efficiency of galaxy formation. We will put constraints on the three parameters from the observed number density and clustering amplitude for each galaxy population. In short, the number density of galaxies is most sensitive to M1 which changes the average number of galaxies per halo. The clustering amplitude on large scales is determined by the hosting halos and thus very sensitive to the mass of those halos, Mmin. The clustering on smaller scales, on the other hand, depends on those three parameters in a fairly complicated fashion; roughly speaking, Mmin changes the amplitude, while α, and to a lesser extent M1 as well, change the slope.

With the above relation, the number density of the corresponding galaxy population at redshift z is given by

∫ ∞ ng,z(z) = dM nhalo(M, z)Ng(M ), (126 ) Mmin
where nhalo(M ) denotes the halo mass function.

The galaxy two-point correlation function on small scales is dominated by contributions of galaxy pairs located in the same halo. For instance, Bullock et al. [8] adopted the mean number of galaxy pairs ⟨Ng (Ng − 1)⟩(M ) within a halo of mass M of the form:

( 2 |{ N g(M ) for Ng (M ) > 1, ⟨Ng (Ng − 1 )⟩(M ) = N 2g(M )log(4Ng (M ))∕log(4) for 1 > Ng (M ) > 0.25, (127 ) |( 0 for Ng (M ) < 0.25.

In the framework of the halo model, the galaxy power spectrum consists of two contributions, one from galaxy pairs located in the same halo (1-halo term) and the other from galaxy pairs located in two different halos (2-halo term):

tot 1h 2h Pg (k) = Pg (k) + Pg (k ). (128 )
The 1-halo term is written as
1 ∫ Pg1h(k ) = ----3-2-- dM nhalo(M )⟨Ng(Ng − 1)⟩b(M )|y (k, M )|p. (129 ) (2π) ng,z
Seljak [77Jump To The Next Citation Point] chose p = 2 for ⟨N (N − 1)⟩ > 1 g g and p = 1 for ⟨N (N − 1)⟩ < 1 g g. The 2-halo term on the assumption of the linear halo bias model [59Jump To The Next Citation Point] reduces to
[∫ ]2 2h Plin(k)- P g (k) = n2g,z dM nhalo(M )Ng (M )b(M )y(k,M ) , (130 )
where Plin(k) is the linear dark matter power spectrum, b(M ) is the halo bias factor, and y(k, M ) is the Fourier transform of the halo dark matter profile normalized by its mass, y(k, M ) = &tidle;ρ(k,M )∕M [77].

The halo occupation formalism, although simple, provides a useful framework in deriving constraints on galaxy formation models from large data sets of the upcoming galaxy redshift surveys. For example, Zehavi et al. [105] used the halo occupation formalism to model departures from a power law in the SDSS galaxy correlation function. They demonstrated that this is due to the transition from a large-scale regime dominated by galaxy pairs in different halos to a small-scale regime dominated by those in the same halo. Magliocchetti and Porciani [47] applied the halo occupation formalism to the 2dFGRS clustering results per spectral type of Madgwick et al. [45Jump To The Next Citation Point]. This provides constraints on the distribution of late-type and early-type galaxies within the dark matter halos of different mass.


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