3. Galaxy evolution
The formalism introduced in the previous section only deals with the formation of dark halos through the action of gravity. To infer the mass function of galaxies we need to precise when such a halo will form a galaxy, that is we have to express explicitly the physical processes which lead to the formation of a galaxy.
3.1. Galaxy formation: cooling constraints
Following the picture of the spherical model, an object with a small overdensity at early times will gradually separate from the general expansion, reach a maximum radius at time and finally collapse at time . One usually assumes that the real halo will virialize at the radius at time , with . We consider that once the halo of dark matter is virialized, the gas can cool and fall into the potential well as it is no longer thermally supported. However, at time on average we assume that another larger halo containing the previous one will eventually form and will merge the initial overdensity with neighbouring ones. Indeed in the usual hierarchical scenarios, which we consider in this article, larger scales turn non-linear later so that small objects get embedded within increasingly large mass condensations. If the gas contained in the small halo has not cooled sufficiently, it will be distributed over the newly formed overdensity. In other words, the gas has only the time interval to cool and fall into the initial halo in which it was embedded in order to allow the latter to retain its individuality and possibly form stars to become a galaxy. Fig. 1 displays this sequence of events. Of course cooling does not stop at time . It may go till the present epoch, and the infall of baryons as well as star formation may still be active at much later time.
Thus, we write this cooling constraint as where is the cooling time. Hence the amount of gas which can cool and fall to the center of the halo, and possibly form a disk and stars, is the mass of gas which is enclosed by the cooling radius where . However, for high redshifts or small temperatures, this radius can be larger than the virialization radius , where the averaged density contrast is equal to (given by p). In this case, the actual extension of the halo where the gas comes from is , since the surroundings have not collapsed yet. Hence we define the radius of the halos we consider by the constraints:
where is the density of the universe at the redshift we consider. The parameter q should be of order unity (we shall take ). Thus, the halos we considered in the definitions (4) and (9) of the mass function will be characterized by an external radius equal to Min. This defines their extension and the function Max introduced in Sect. 2. As explained in Appendix A we do not need to assume a smooth power-law density profile for dark matter halos. Indeed, objects defined by are not defined as being the central core of radius within a larger halo of radius assumed to have a power-law density profile. We directly consider mass condensations satisfying the criteria (15) without reference to the properties of a possible larger object. The advantage of this procedure is that clusters automatically contain several galaxies (as they should) instead of only one central huge galaxy , see also the discussion below in Sect. 3.2. The cooling time is given by:
where is the cooling function (in ). We use the cooling function given by Sutherland & Dopita (1993) for a gas with primordial abundances. The expression (16) for is the evaluation for the mean halo baryonic density. Cooling may be much more efficient since the local density increases within the halo and the baryon density gets higher during the collapse. This means that the parameter s which enters (16) should be larger than unity. A crude estimate for s using a mean power-law density profile (with no allowance for the cooling to be increased due to baryon concentration that would lead to even larger a value: such an increase has been shown to exist in numerical simulations, but is naturally limited, see Teyssier et al. 1998) shows it may indeed reach several units.
It is convenient to express (15) as a relation virial temperature - density, so that the halos we consider can be completely described by their temperature. Hence we can write the curve corresponding to the first condition of the system (15) as:
The curve corresponding to the second condition of (15), which characterizes just-virialized objects, is given by:
where is the density contrast at the time of virialization, at the redshift z we consider, and is the mean density of the universe at that redshift. If this density contrast is a constant: . The general behaviour of these curves is shown in Fig. 2. The curve which describes the system (15) is simply given by:
This relation temperature-density defines implicitely the relation we introduced in Sect. 2 to determine the mass function of galaxies. Note that there are only two parameters: p and . From the spherical model we choose as a natural value, while the product will be given by the luminosity of the Milky Way (which implies a well defined mass of baryons), with the constraint that both s and q should be of order of a few units.
3.2. Galaxies versus clusters and groups
The curves and are shown in Fig. 2. Objects below either one of these curves violate one at least of the above conditions. Objects above these curves satisfy the criteria, but it is easy to see that slightly larger size objects (and whence lower density objects) also satisfy the criteria. Hence we define galaxies as being, for a given temperature (velocity dispersion), the largest object (hence the one with the lowest density) that satisfies the criteria (15). Thus the locus of galaxy formation, curve , is the higher of the curves and in the -T diagram. Objects lying below this curve will be groups of galaxies or galaxy clusters. These objects do not cool in a uniform fashion within a Hubble time at formation, but may have cooled by now (groups) or are still cooling (X-ray clusters). The consideration of the latter in some sense justifies our cooling requirement for galaxies: we clearly see clusters cooling at the present epoch but they do not constitute a single galaxy. However, cooling is known to be necessary for galaxies to form and we would argue that the above condition is the most natural one to avoid the baryonic component of the following merging event to mix completely. Groups and clusters contain some of the galaxies we are considering in the present paper and they can be described in a consistent way by means of the same methods developed in VS and used here to describe the galaxy distribution. Their multiplicity, counting the same objects that the ones we deal with here but grouped differently, will be calculated elsewhere (Valageas & Schaeffer 1998). In particular, the normalization conditions (7) and (11) show that if we integrate from up to we recover the total mass of the universe whatever the curve . The latter only describes how one can divide the matter content of the universe and different choices simply correspond to different classifications: one can count a group as a single large object or as the assembly of several distinct galaxies which are considered as individual entities. In this article we adopt this latter point of view since we are interested in galaxies themselves. In fact, the galaxy mass function does not extend down to because halos with a low virial temperature K do not cool (due to inefficient cooling as well as heating by the UV background). These patches of matter (which fill most of the volume of the universe with voids) form the Lyman- forest clouds which we describe in details in Valageas et al. (1999b).
The curve does not depend on redshift while the curve does. At fixed temperature T, as defined by increases with z along with the average density of the universe. Whence the locus of galaxy formation varies with z and at early times it becomes identical to .
This variation with redshift has important consequences. Galaxies, that at a given epoch settle on the curve because larger objects of lower densities are not virialized, at a later epoch will lie above the curve and whence according to our criteria will be embedded within larger objects. This provides for these galaxies a continuous merging process with obviously an evolution of the mass as well as the number of such objects (and indeed an associated star formation). On the other hand, galaxies that settle on the curve at a later time will be limited by the same condition on density: larger objects of lower density will not satisfy the above conditions and thus may be groups or galaxy clusters, but cannot become a larger galaxy into which the galaxy we consider gets embedded. So in the framework of our model these galaxies will no longer evolve neither in mass nor in number. We call these galaxies "isolated galaxies". Thus we have a population of smaller galaxies that evolve by mergers and a population of larger galaxies that bear only internal evolution. At early times all objects are continuously merging since the only limiting curve at high z is . As time goes on, the larger objects get limited by cooling, settling on . So all these now "quiet galaxies" have an early phase with strong merging processes. Once this early phase is over, the now quietly evolving galaxy may for instance form a disk. Within this picture, we do not take into account the strong merging processes that take place in the very dense cores of clusters. Such additional interactions may sufficiently disturb (Balland et al. 1998) the galaxies we model here to change them into ellipticals, a plausible explanation of the density morphology relation. This will be accounted for in a subsequent paper about galaxy clusters (Valageas & Schaeffer 1998).
Note that in our model all galaxies have roughly the same age (of the order of the age of the universe) from the star formation viewpoint, even if they did not exist as distinct objects along this whole period. Indeed, present galaxies are the result of the merging of many smaller and older sub-units where star formation was already active at high z. In fact, contrary to the usual statement, although we work within the framework of the standard hierarchical clustering scenario massive galaxies look slightly older than small ones (see Appendix C). Because of their more efficient star formation process (due to their higher density and virial temperature, see Appendix B) massive and bright galaxies have a redder and older stellar population.
3.3. Star formation
Our scenario describing the history of star formation is rather standard, and kept as simple as possible. Apart from a small component (10%) of dark baryonic matter (brown dwarfs, planets) that plays a negligible role in the stellar evolution history, we consider four components:
The star formation rate is proportional to the mass of central gas with a time-scale proportional to the dynamical time-scale :
The mass of gas heated and ejected by supernovae out of the central parts of the galaxy is proportional to the star formation rate and decreases for deep potential wells:
where is a constant and T the halo virial temperature, see (B2). Finally, the infall of gas from to occurs on a dynamical time-scale and not on the cooling time-scale since because of (15):
Since we shall use:
where and are parameters of order unity. We assumed , as it may describe gravitational instabilities within galaxies as well as the influence of neighbours. Moreover, the time-scale disappears for faint galaxies, as we shall see below, and our model would still be approximately valid even for bright galaxies if one has in fact , since it would simply correspond to a change of (then the system is governed by the longest time-scale among and ). This means that other forms for , for instance or (where is the gas density in the core) would give similar results, as we checked numerically.
The details of this model of star formation are presented in Appendix B and Appendix C. We solve the corresponding equations (B1) numerically but we also give there approximate analytic solutions that allow one to follow the behaviour of these various components as a function of time and galactic mass. We can stress here that the fraction of non-luminous stars we adopt (10%) is consistent with observations (see discussion in Mera et al. 1998) while the values used in some other studies (50% in Kauffmann et al. 1993; 63% in Cole et al. 1994 for instance) are much too large. Using a smaller fraction of non-luminous stellar-like objects however would increase the luminosity of the galaxies obtained in the latter models and would lead to stronger discrepancies with the observed luminosity function.
We also describe the evolution of the metallicity (E1) with three different values: for stars, for the central gaseous component and for the diffuse gaseous component.
We summarize here the free parameters of our model:
The parameters q, and must be of order unity and they are constrained by the luminosity, mass, gas/star mass ratio of the Milky Way, while is constrained by the supernovae rate of similar galaxies. The yield y is given by the metallicity of the solar neighbourhood. Thus all parameters of our model are adjusted on other observations than those related to the luminosity function we seek to reproduce. Note moreover that for bright and massive galaxies () is irrelevant: the effect of supernovae is negligible since the potential well is very deep so that gas cannot be ejected efficiently. For faint galaxies () is irrelevant because a quasi-stationary regime sets in very quickly where the galaxy evolution is governed by the equilibrium between the ejection of central gas by supernovae and the infall from the diffuse component (see B.2) so that the star formation rate is given by:
where is the total gas mass. For massive galaxies the derivation leading to this relation is no longer valid but (24) still gives a reasonable description of the star formation rate because in our model so that for there is only one time-scale which is also given by since in this case . The value of the main parameters and is further discussed in Appendix F.
© European Southern Observatory (ESO) 1999
Online publication: April 19, 1999