3. Model predictions for the LBG population
Since the cooling regulation discussed above gives specific predictions of how star formation may have proceeded in LBGs, here we use this model to predict the properties of the LBG population. The condition in Eq. (14) implies that the star formation rate in a disk is equal to the rate of gas infall (due to a balance between cooling and heating). Thus the evolution of the gas in the disk of an LBG host halo is described by the standard chemical evolution model with infall rate equal to star formation rate, i.e., the new infalling gas to the disk distributed radially in an exponential form with the scale length of , and the reheated gas removed decreases with the increasing radius due to the decreasing SFR. Under the instantaneous recycling approximation (Tinsley 1980), the gas metallicity Z is given by
where is the initial metallicity of the infalling gas, y is the stellar chemical yield, is the gas surface density (which is kept constant by gas infall) and is the total mass surface density, which increases as star formation proceeds:
Here the enrichment of the halo hot gases is not taken into account because the amount of metals heated up to the haloes by SNs is relatively smaller than that of primordial gases.
3.1. Individual objects
Fig. 2 shows the star formation rate as a function of halo circular velocity and spin parameter . As expected, the predicted SFR increases with but decreases with . As we can see from the figure, if we define systems with (which matches the SFRs for the observed LBG population) to be LBGs, the majority of their host haloes must have which are cooling dominated. This result is the same as that obtained by MMWb based on the observed number density and clustering of LBGs. Thus, the star formation rate based on cooling argument is also consistent with the observed number density and clustering. Because SFR is higher in a system with smaller , the LBG population are biased towards haloes with small spins, but given its relatively narrow distribution, this bias is not very strong.
The predicted metallicity gradients on individual disks are shown in Fig. 3 for two different choices of star formation time scale of 0.5Gyr and 1Gyr respectively, where we assume that and in order to make the predictions easily comparable with observations. The metallicity gradients are negative in all cases. When radius is measured in disk scale length, the predicted metallicity depends weakly on but strongly on , and is higher for a longer star formation time. As one can see from Eq. (15), the largest metallicity in the model is . This metallicity can be achieved in the inner part of compact disks (with small ) when star formation time Gyr. The metallicity drops by a factor of from its central value at .
3.2. LBG population
Since the distribution of haloes with respect to and are known, we can generate Monte-Carlo samples of the halo distribution in the - plane at any given redshift. We can then use the galaxy formation model (MMWb) discussed above to transform the halo population into an LBG population based on LBGs with highest SFRs which is the same as that outlined in Sect. 2.
We define the typical metallicity of a galaxy as the one at its effective radius. Fig. 4 shows the distribution of this metallicity for two choices of the star formation time, Gyr and 1 Gyr. Just as the same reason as Fig. 3 in last section, we have assumed that and in order to make the predictions easily comparable with observations. The median values of are 0.60 and 0.84 for Gyr and 1 Gyr, respectively. The sharp truncation at is due to the fact that this quantity has a maximum value of 1 in the present chemical evolution model. It can be inferred from Fig. 3 that the range in decreases with increasing star formation time. Thus, if gas infall lasts for a long enough time, the distribution in will be very narrow near 1 and all LBGs will have metallicity . According to the works of Tinsley (1980) and Maeder (1992), the stellar yield y is of the order of for the Salpeter IMF. If we adopt a stellar yield and , and if LBGs are not short bursts (e.g. Gyr) then their metallicity will be which is similar to that proposed by Pettini (1999).
The predicted distribution of effective radii for the LBG population is shown in Fig. 5. The distribution is similar to that of MMWb. The predicted range is with a median value of 2.5 kpc. Note that the effective radii in the cooling-regulated model are independent of the star formation time and . The model prediction is in agreement with the observational results given by Pettini et al. (1998), Lowenthal et al. (1997) and Giavalisco et al. (1996) which are mentioned above.
The predicted SFR distribution of LBGs also resembles the prediction of MMWb except for a slight difference with MMWb, which is shown in Fig. 6. The median values are 180 for the model and spans from 100 to 500. To compare with observations, we have to take into account the effect of dust. If we apply an average factor of 3 in dust extinction, then the predictions closely match the values derived from infrared observations by Pettini et al. (1998) although there might exist rare LBGs with very high SFR.
3.3. Contribution to the soft X-ray and UV background
Since the virial temperature of LBG haloes is quite high, in the range of K, significant soft X-ray and hard UV photons may be emitted as the halo hot gas cools. It is therefore interesting to examine whether the LBG population can make substantial contribution to the soft X-ray and UV backgrounds.
The dominant cooling mechanism for hot gas with temperature K is the thermal bremsstrahlung. The bremsstrahlung emissivity is given by (e.g., Peebles 1993)
where (in ) is the electron density and T (in K) is the temperature given by Eq. (6). The total power emitted per unit volume is
We write the total luminosity in thermal bremsstrahlung as
and we take here as WF so that is equal to the initial thermal energy in the cooling gas. Note that the value of is quite uncertain because it depends on the density and temperature profiles of the hot gas. Substituting Eq. (13) into the above equation, we obtain the total soft X-ray luminosity for an LBG
is the fraction of total energy that falls into the ROSAT soft X-ray (0.5-2 keV) band. The contribution of the LBG population to the soft X-ray background is then
where is the comoving number density of LBG haloes as a function of redshift z, is the differential comoving volume from z to and is the luminosity distance. The integrate for is to sum up all selected LBGs with based on their highest SFRs. We have integrated over redshift range from 3 to 4 where the number density of LBGs is nearly a constant (Steidel 1999a,b). This contribution should be compared with the value derived from the ROSAT observations (Hasinger et al. 1993) in the 0.5-2 keV band
As we can see, the soft X-ray contribution from LBGs could be a substantial fraction (about 20%) of the total soft X-ray background.
Similarly we can calculate the contribution of LBGs to the UV background at . We evaluate the UV background at 4 Ryd (1Ryd=13.6 eV) using nearly identical procedures, we find that
which is much smaller than the UV background from AGNs, (e.g. Miralda-Escude & Ostriker 1990).
3.4. Contribution to the total metals
Based on the recent observational results of the cosmic star formation history, Pettini (1999) obtained a predicted total mass of metals produced at . After combining results of all contributors observed, he argued that there seems to exist a very serious "missing metal" problem, i.e., the predicted result is much higher than the observed ones. So, it is interesting to evaluate the total metals produced by LBGs in our model.
According to the method we select LBGs to be the galaxies with highest SFR and our chemical evolution model mentioned in Sect. 3.2, we can calculate the total metal density produced by the LBG population at based on their observed comoving number density which is for the assumed cosmology (Adelberger et al. 1998). Defining that is the metal density relative to the critical density, we get that of LBGs are and for star formation time of 0.5Gyr and 1Gyr respectively, where y is the stellar yield which is the same as above. Because the virial temperature of LBG haloes are very high, a significant fraction of the metal should be in hot phase. Comparing our results with that estimated by Pettini (1999) which is (the cosmogony has been taken into account), we find that there is no "missing metal" problem in our model.
3.5. LBGs and damped Lyman-alpha systems
Damped Lyman-alpha systems (DLSs) are another population of objects that can be observed at similar redshift to LBGs. The DLSs are selected according to their high neutral HI column density (), and are believed to be either high-redshift thick disk galaxies (Prochaska & Wolfe 1998) or merging protogalactic clumps (Haehnelt et al. 1998). In either case, to match the observed abundance of DLSs, most DLSs should have circular velocity between to , much smaller than the median circular velocity of LBGs (). Based on the PS formalism (Eq. (5)) and disk galaxy formation scenario suggested by MMWa (Eqs. (1) and (2)), we can estimate with the random inclination being taken into account, that the fraction of absorbing cross-sections contributed by LBGs amounts to only about 5% of the total absorption cross-section assumed LBGs with highest SFRs. This means that only a very small fraction of DLSs can be identified as LBGs.
The physical connection between LBGs and DLSs is still unclear, although the recent observation of Moller & Warren (1998) using HST indicates that some DLSs could be associated with LBGs. In Fig. 7, we show the predicted metallicity distribution for the subset of DLSs which can be observed as LBGs. Again, we have assumed that and to make the predictions more easily comparable to observations. As can be seen, the DLSs generally have lower metallicity than LBGs, because they are biased towards the outer region of the host galaxies, where the star formation activity is reduced. Note, however, that the metallicity of these DLSs could still be higher than most DLSs at the same redshift, which typically have a metallicity of (Pettini et al. 1997a).
© European Southern Observatory (ESO) 2000
Online publication: February 25, 2000