and WMAP3 (red) simulations. The significantly lower value of σ8 in the WMAP3 case reduces the overall normalisation. In addition, the red tilt in the primordial power spectrum index makes the difference greater on small scales than on large scales. Indeed, on the scales responsible for the low ℓmicrowave background fluctuations the two power spectra are almost the same, reflecting the fact that they are both required to fit observed CMB fluctuations on these scales. Conversely, the suppression of power is particularly significant on the small scales responsible for the formation of the first nonlinear structures.
4.3 The Semi-Analytic Model
In this paper we use the galaxy formation model described in De Lucia & Blaizot (2007) which is based on the WMAP1 cosmology and builds on previous work by the “Munich” galaxy formation group (Kauffmann et al. 1999a; Springel et al. 2001; De Lucia et al. 2004; Springel et al. 2005; Croton & et. al 2006). Although not in perfect agreement with all as- pects of the local galaxy population (see, for example, Weinmann et al. (2006)), this model does quite a good job of reproducing the observed relations between stellar mass, gas mass, and metallicity (De Lucia et al. 2004), the observed luminosity, colour, and morphology distributions (Croton & et. al 2006; De Lucia et al. 2006) and the observed two-point cor- relation functions (Springel et al. 2005). Kitzbichler & White (2007) have recently shown that it also agrees reasonably well with the observed galaxy luminosity and mass functions at high redshift. We refer the reader to the original papers for a full description of the numerical implementation, and of the physical processes modelled. In the following, we summarise briefly the treatment of those physical processes for which we needed to change the efficiency parameters in order to maintain agreement with observations of the local galaxy population when we switch from WMAP1 to WMAP3 (see Table 4.3 and Sec. 4.5). In the semi-analytic model we use in this work, star formation is assumed to occur at a rate given by:
˙
m∗=αSF(mcold−mcrit)/tdyn,disc (4.1) where mcold and tdyn,disc are the cold gas mass and the dynamical time of the galaxy, defined as the ratio between the disk radius and the virial velocity, mcrit corresponds to a critical value for the gas surface density (Kennicutt 1998; Kauffmann 1996; Mo et al. 1998), and αSF controls the efficiency of the transformation of cold gas into stars when the gas surface density is above the critical value. (See Croton & et. al (2006) for more detailed descriptions of the implementation of this process and of the feedback processes described below.)
Massive stars explode as supernovae shortly after a star formation event and are as- sumed to reheat a gas mass that is proportional to the mass of stars formed (based on the observations of Martin 1999):
∆mreheated=ǫdisk∆m∗, (4.2)
Again following Croton & et. al (2006), we write the energy released by an event which forms a mass ∆m∗ in stars as:
where 0.5V2
SN is the mean supernova energy injected per unit mass of newly formed stars, andǫhalo represents the efficiency with which this energy is able to convert cold interstellar medium into hot, diffuse halo gas. The amount of gas that leaves the dark matter halo in a “super-wind” is determined by computing whether excess SN energy is available to drive the flow after reheating of material to the halo virial temperature.
As in Kauffmann & Haehnelt (2000), black holes are formed and fuelled during mergers: ∆mBH= f′ BHmcold 1 + (280kms−1/V vir)2 (4.4) We assume here that black holes grow during both major and minor mergers, and that the efficiency of gas accretion onto the black hole scales with the baryonic mass ratio of the merging galaxies:
fBH′ =fBH×(msat/mcentral) (4.5) This is the primary process driving the growth of the total mass in supermassive black holes. Individual black holes can also gain mass through merging when their host galaxies merge.
Finally, we use the model by Croton & et. al (2006) to describe heating by centrally located AGN in massive groups and clusters. This process is assumed to be associated with “radio mode” outflows which suppress cooling flows and thus the condensation of gas onto the central galaxies. The process is assumed to occur whenever a massive black hole finds itself at the centre of a static hot gas halo. During this phase, the accretion rate onto the central supermassive black hole is taken to be:
˙ mBH,R =κAGN mBH 108M ⊙ fhot 0.1 Vvir 200kms−1 3 , (4.6)
wheremBH is the black hole mass,fhot is the fraction of the total halo mass in the form of hot gas, Vvir is the virial velocity of the halo, andκAGN is efficiency parameter with units ofM⊙yr−1. The energy released during this accretion process is used to reduce the cooling
flow. Croton & et. al (2006) showed that this results in complete suppression of cooling in relatively massive haloes and groups. The process starts being effective at a mass scale that evolves as a function of redshift as shown in their Fig. 7.
In our simulation scheme, haloes (and the galaxies within them) are followed even after they are accreted onto larger systems. The dynamics of such a satellite subhalo is fol- lowed explicitly by the N-body simulation until tidal stripping causes its mass to fall below the resolution limit of the simulation (Ghigna et al. 2000; De Lucia et al. 2004; Gao et al. 2004). When this happens, we estimate a survival time (tmerge) for the associated galaxy using its current orbit and the classical dynamical friction formula of Binney & Tremaine (1987). Once this time has elapsed, the galaxy is assumed to merge onto the central galaxy of its current halo. While it still survives it is assumed to follow the particle which was the most bound particle of the subhalo at the last time it was identified. De Lucia & Blaizot (2007) found that increasing the merging time by a factor of 2 slightly improves the fit to observed luminosity function. Such an increase has other effects which De Lucia & Blaizot (2007) did not study, for example, it increases the amplitude of small-scale galaxy cor- relations by about a factor of 2 at r < 100 kpc. Others authors have claimed that this