3. Galaxy formation
In this paper we wish to study the reionization history of the universe. Since a large part of the ionizing radiation will be emitted by stars, we first need to devise a model for galaxy and star formation. We shall use a simplified version of the model described in detail in VS II and which was there compared with many observations. One can define galaxies by the requirement that two constraints be satisfied by the underlying dark matter halo: 1) a virialization condition (where is given by the spherical model and is constant for a critical universe) and 2) a cooling constraint which states that the gas must have been able to cool within a few Hubble times at formation. However, at high redshifts the cooling constraint becomes irrelevant since any object which satisfies 1) also satisfies 2). Hence since we are mainly interested in large redshifts we shall simply define galaxies by the virialization condition . We also require that the virial temperature T of the halo be larger than the "cooling temperature" at redshift z. The latter corresponds to the smallest virialized objects which can cool efficiently at redshift z, defined by the constraint:
where is a proportionality factor (one must have since cooling is more efficient within the halo where the density is larger than on its boundary and cooling accelerates as baryons collapse). Here is the age of the universe at redshift z while is the cooling time of a halo with density contrast , mass M, taking into account both cooling (recombination, molecular cooling) and heating (by the background UV flux) processes. Since the physical properties of virialized halos with temperature T and density contrast are different from the IGM, we let the chemical reactions (involving HI, HII, H-, H2, H, HeI, HeII, HeIII and e-) evolve for a Hubble time within this environment (defined by T and ) before we evaluate the cooling time . The main effect is that at large redshifts such clouds may produce enough molecular hydrogen to make molecular cooling efficient while with the use of the IGM abundances one would underestimate this contribution; see for instance Tegmark et al. (1997) for a detailed discussion. This will also appear clearly below in Fig. 4 where we compare the main contributions to cooling for both the IGM and these cooling halos. The virial temperature also defines the mass and the radius of the smallest objects which can cool and eventually form stars at redshift z. From the lower-bound and the virialization constraint , we obtain the mass function of galaxies at redshift z using (2).
Next, we must attach a specific stellar content to these galactic halos. We shall again use the star formation model described in VS II. This involves 4 components: (1) short-lived stars which are recycled, (2) long-lived stars which are not recycled, (3) a central gaseous component which is deplenished by star formation and ejection by supernovae winds, and replenished by infall from (4) a diffuse gaseous component. The star-formation rate is proportional to the mass of central gas with a time-scale set by the dynamical time. The mass of gas ejected by supernovae is proportional to the star-formation rate and decreases for deep potential wells as , in a fashion similar to that adopted by Kauffmann et al. (1993). It was seen in VS II that for such a model a good approximation for the star-formation rate is:
Here is the fraction of the energy delivered by supernovae transmitted to the gas ( erg) while is the number of supernovae per solar mass of stars formed. Note that for halos defined by a constant density threshold we have . Although (6) was obtained for small galaxies with (which is the range we are mainly interested in) it also provides a reasonable approximation for large galaxies . In the case we obtain in our model for a galaxy similar to the Milky Way (i.e. with a circular velocity km s-1): years and year (see VS II). This star formation rate is consistent with observations (McKee 1989). Then the mass of gas at time t within the galaxy is given by:
where is the initial mass of baryons which we take to be proportional to the dark matter mass M:
where is a parameter of order unity which enters the definition of the dynamical time . The significance of each term in this expression is clear and the temperature dependence simply states that the average star formation efficiency of small galaxies is small as the gas is easily expelled by supernovae. Note that in the original model described in VS II for bright galaxies at low redshifts, the star formation rate declines since most of the gas has already been consumed. This does not appear in (10) because we defined all galaxies by while at low z for large T the cooling constraint implies that (i.e. R is constant) which decreases the galactic dynamical time and increases the ratio which enters (8).
To derive the radiation emitted by galaxies, we do not need their global star formation rate but their stellar content. However, as shown in VS II, the mass in the form of short-lived stars (i.e. with a life-time small compared to ) of mass m to is given by:
where is the fraction of mass which goes into such stars for each unit mass of stars which are formed. This depends on the initial stellar mass function (IMF) . Since the stellar radiation output at high energy ( eV) is dominated by the most massive stars, the relation (11) will be sufficient for our purposes. Next, if we assume that stars radiate as blackbodies with an effective temperature and we use the mean scalings and we obtain the energy output of such galaxies:
From the radiation emitted by individual galaxies we now wish to estimate the energy received by a random point in the IGM. We shall write the source term due to stellar radiation for the background UV flux , see (30), as the following average:
where is the mass function of galaxies, obtained from (2) as described previously, while is a mean opacity which takes into account the fact that the radiation emitted by galaxies can be absorbed by the IGM and Lyman- clouds. We shall come back to this term later. Thus, we get in this way a simple model for the stellar radiative output from our more detailed description of galaxy formation. The reader is referred to VS II for a more precise account of the details and predictions of our galaxy formation model. Note that our prescription is consistent with such observations as the Tully-Fisher relation and the B-band luminosity function.
© European Southern Observatory (ESO) 1999
Online publication: June 18, 1999