2.1. Galaxy formation
In this paper, we use the galaxy formation scenario described in Mo et al. (1998a, hereafter MMWa) to model the LBG population. In this scenario, central galaxies are assumed to form in dark matter haloes when collapse of protogalactic gas is halted either by its angular momentum, or by fragmentation as it becomes self-gravitating (see Mo et al. 1998b, hereafter MMWb, for details). As described in MMWb, the observed properties of LBGs can be well reproduced if they are assumed to be the central galaxies formed in the most massive haloes with relatively small spins at . As in MMWb, we assume that gas in a dark halo initially settles into a disk with exponential surface density profile.
When the collapsing gas is arrested by its spin, the central gas surface density and the scale length of an exponential disk are
where is the fraction of halo mass that settles into the disk, is the circular velocity of the halo, is the dimensionless spin parameter, is the Hubble constant at redshift z and is its present value (see MMWa for details). Since increases with z, for a given disks are less massive and smaller but have a higher surface density at higher redshift. When is low and is high, the collapsing gas will become self-gravitating and fragment to form stars before it settles into a rotationally supported disk. In this case, we will take an effective spin in calculating and .
We take the empirical law (Kennicutt 1998) of star formation rate (SFR) to model the star formation in high-redshift disks which is
respectively. Here is the SFR per unit area and is the gas surface density. Note that this star formation law was derived by averaging the star formation rate and cold gas density over large areas on spiral disks and over starburst regions (Kennicutt 1998). We will apply this law differentially on a disk and also take into account the Toomre instability criterion of star formation (Toomre 1964; see also Binney & Tremaine 1987).
For a given cosmogonic model, the mass function for dark matter haloes at redshift z can be estimated from the Press-Schechter formalism (Press & Schechter 1974):
where with being the linear growth factor at z and , is the linear rms mass fluctuation in top-hat windows of radius R which is related to the halo mass M by , with being the mean mass density of the universe at . The halo mass M is related to halo circular velocity by . A detailed description of the PS formalism and the related cosmogonic issues can be found in the Appendix of MMWa.
From the Press-Schechter formalism and the -distribution which is a log-normal function with mean and dispersion (see Eq.  in MMWa), we can generate Monte Carlo samples of the halo distributions in the - plane at a given redshift and, using the star formation law outlined above, assign a star formation rate to each halo. As in MMWb, we select LBGs as the galaxies with the highest star formation rate, so that the comoving number density for LBGs is equal to the observed value, for the assumed cosmology at , as given in Adelberger et al. (1998). Here it is worth noting that the model selection of LBGs we adopted is without the dust extinction being considered. This implies that the contribution of the dust is assumed to be uniform. But in fact, it could be very different from galaxies to galaxies. So, our selection of LBGs may not have one-to-one correspondence with the observed LBGs (Baugh et al. 1999), but the selection should be correct on average.
2.2. Cooling-regulated star formation
What regulates the amount of star-forming gas in a dark halo? In the standard hierarchical scenario of galaxy formation (e.g. White & Rees 1978; White & Frenk 1991, hereafter WF), gas in a dark matter halo is assumed to be shock heated to the virial temperature,
as the halo collapses and virializes. The hot gas then cools and settles into the halo centre to form stars. As suggested in WF, the amount of cold gas available for star formation in a dark halo is either limited by gas infall or by gas cooling, depending on the mass of the halo. For the massive haloes () we are interested here, gas cooling rate is smaller than gas-infall rate, and the supply of star-forming gas is limited by gas cooling (see WF for details). It is therefore likely that gas cooling is the main process that constantly regulates the SFR in LBGs.
where is the current gas content of the disk. The rate at which gas is consumed by star formation is therefore
where is the returned fraction of stellar mass into the ISM; we take for a Salpeter IMF (e.g. Madau et al. 1998). According to WF, the heating rate due to supernova explosions under the approximation of instantaneous recycling can be written as
where is an efficiency parameter which is still very uncertain. We take it to be 0.02 as in WF. The rate at which gas is heated up (to the virial temperature) is therefore
Comparing this equation with Eqs. (7) and (8), we can find that the rate for gas consumption due to star formation is much larger than the rate of gas heating for LBG haloes. Because LBGs are hosted by massive haloes which have large circular velocities , the haloes are cooling dominated which is confirmed during the detailed calculation below. Following WF we define a mass cooling rate by
where is the cooling radius and is the density profile of the hot gas in the halo. For simplicity, we assume that , and we define to be the radius at which the cooling time is equal to the age of the universe, which is similar to the time interval between major mergers of haloes (Lacey & Cole 1994). The density distribution of the halo mass here is assumed to be isothermal. However, it is the NFW profile (Navarro et al. 1997) in MMWb. Because the difference of the resulting cooling rates between these two different choices of density profiles is small (Zhao et al. 1999), and since the major goal here is to show whether or not the cooling-regulated star formation can be valid, the adoption of isothermal profile will not influence the final result very much.
Under this definition, gas within the cooling radius can cool effectively before the halo merges into a larger system where it may be heated up to the new virial temperature if it is not converted into stars. Using the cooling function given by Binney & Tremaine (1987) where cooling function in the range of (and assuming gas with primordial composition), the mass cooling rate can then be written as
If is smaller than , then cold gas will accumulate in the halo centre and lead to higher star formation rate. If, on the other hand, , the amount of cold gas will be reduced by star formation and supernova heating, leading to a lower star formation rate. We therefore assume that there is a rough balance among these three rates:
It should be noted that the cooling-regulated star formation process is only a reasonable hypothesis, and the real situation must be much more complicated. For example, during a major merging of galactic haloes, the amount of gas that can cool must be much larger than that given by the cooling argument, and the star formation may occur in a short burst (e.g. Mihos & Hernquist 1996). However, such bursts are not expected to dominate the observed LBG population, because of their brief lifetimes. Thus, star formation rates in the majority of LBGs are expected to be regulated by Eq. (14) on average. As shown in MMWb, to match the observed number density of LBGs, the median value of is about in the present cosmogony. The typical star formation rate is of the order of . This is not very different from the observed star formation rates, albeit dust distinction in the observations may be difficult to quantify.
Fig. 1 shows the value of required by the balance condition Eq. (14) as a function of halo circular velocity, assuming that and the left hand side exactly equals the right hand one in Eq. (14). Results are shown for two choices of spin parameters, and 0.08, corresponding to the 50 and 90 percent points of the distribution for the LBG population (MMWb). As one can see, for the majority of LBG hosts, gas cooling indeed regulates the values of to the range from 0.02 to 0.04. So, we can reasonably choose for the LBG population as MMWb did. Since the cooling time is approximately the age of the universe at , cooling regulation ensures that star formation at the predicted rate can last over a large portion of a Hubble time.
© European Southern Observatory (ESO) 2000
Online publication: February 25, 2000