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Astron. Astrophys. 350, 725-742 (1999)

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2. Model

First, we briefly present the main characteristics of our model. Some of them are described in more details in Valageas & Silk (1999a) (hereafter VS).

2.1. Multiplicity functions

In order to evaluate the reheating and reionization of the universe by stars and quasars we need the mass functions of galaxies and QSOs. We also derive the multiplicity function of Lyman-[FORMULA] absorbers which provide most of the opacity at low z after reionization. This is necessary in order to model the density fluctuations in the IGM itself. Indeed, at low z we consider density contrasts from [FORMULA] (voids) up to [FORMULA] (massive Lyman-[FORMULA] forest clouds) within the IGM. Of course, the need to describe such a large class of objects, defined by various density thresholds (which will also be required by entropy considerations, as shown below) or other constraints (e.g. on their size), means that we cannot use the familiar Press-Schechter prescription (Press & Schechter 1974). Indeed, the latter is restricted to "just-virialized" halos defined by a density contrast [FORMULA]. Thus, we assume that the non-linear density field is well described by the scaling model developed in Balian & Schaeffer (1989). This description is based on the assumption that the many-body correlation functions obey specific scaling laws which can also be seen as a consequence of the stable-clustering ansatz (Peebles 1980):

[EQUATION]

where [FORMULA] is the local slope of the two-point correlation function. This model has been checked through the counts-in-cells statistics against various numerical simulations (e.g. Colombi et al. 1997; Valageas et al. 1999b; Munshi et al. 1999). Then, for a given class of objects defined by a relation [FORMULA] (which implies a specific radius [FORMULA]) we attach to each object the parameter x defined by:

[EQUATION]

where

[EQUATION]

is the average of the two-body correlation function [FORMULA] over a spherical cell of radius R and provides the measure of the density fluctuations in such a cell. Then, we write the multiplicity function of these objects (defined by the constraint [FORMULA]) as (Valageas & Schaeffer 1997):

[EQUATION]

where [FORMULA] is the mean density of the universe at redshift z, while the mass fraction in halos of mass between M and [FORMULA] is:

[EQUATION]

The scaling function [FORMULA] only depends on the initial spectrum of the density fluctuations and must be obtained from numerical simulations. In practice, we use the scaling function [FORMULA] obtained by Bouchet et al.(1991) for a CDM universe. The mass functions obtained in this way have been checked against the results of numerical simulations for various constraints [FORMULA] in the case of a critical universe with an initial power-law power-spectrum (Valageas et al. 1999b). This description also takes into account the substructures which may exist within larger objects and it allows one to derive for instance the amplitude of the density fluctuations in the IGM (i.e. the "clumping factor"). On the other hand, note that a simpler model where the density field is described as a collection of smooth halos with a universal density profile is inconsistent with the results of numerical simulations, as shown in Valageas (1999) (it implies a wrong behaviour of the many-body correlation functions).

2.2. Galaxies

Along the lines of VS, we use a simplified version of the model described in Valageas & Schaeffer (1999a). We define galaxies by two constraints: 1) a virialization condition [FORMULA] (where [FORMULA] is given by the usual spherical model) and 2) a cooling condition [FORMULA] which states that the gas must have been able to cool within a few Hubble times [FORMULA] at formation in order to fall into the dark matter potential well and form a galaxy (see also Rees & Ostriker 1977; Silk 1977). At late times, the second condition 2) is the most restrictive for just-virialized halos (defined by [FORMULA]) with a large virial temperature T. This means that these objects are groups or clusters which contain several subunits (i.e. galaxies) which satisfy the constraint 2) which approximately translates in this case into a constant "cooling radius" [FORMULA] kpc (see Valageas & Schaeffer 1999a for a detailed discussion). Thus, we define galaxies by the relation [FORMULA] such that [FORMULA] except for high temperature halos at low z where this constraint would imply [FORMULA]. Then, these objects are defined by the condition [FORMULA] which gives their density contrast [FORMULA]. This allows us to distinguish clusters or groups from galaxies. In practice, as seen in Valageas & Schaeffer (1999a), the condition 2) only plays a role at low redshift [FORMULA] for high temperature galaxies ([FORMULA] K). Finally, we also require the virial temperature T of the galactic halos to be larger than a "cooling temperature" [FORMULA]. The latter corresponds to the smallest just-virialized objects which can cool efficiently at redshift z, defined by the constraint [FORMULA]. Moreover, it is constrained to be larger than or equal to the temperature of the IGM. At low z we have [FORMULA] K in our original model (VS) since for smaller temperatures cooling is very inefficient (due to recombination). From the lower bound [FORMULA] and the density threshold [FORMULA] we obtain the galaxy mass function using (3).

Then, we use a simple star formation model to derive the stellar content and the luminosity of these galaxies. 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, replenished by infall from (4) a diffuse gaseous component. The star formation rate [FORMULA] 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 [FORMULA], in a fashion similar to Kauffmann et al.(1993). Some predictions of this model (galaxy luminosity function, Tully-Fisher relation) have already been checked against observations (Valageas & Schaeffer 1999a). Thus, we obtain for the mean star-formation rate per Mpc3:

[EQUATION]

with:

[EQUATION]

where [FORMULA] is a parameter of order unity which enters the definition of the dynamical time, while [FORMULA] K describes the ejection of gas by supernovae and stellar winds (see also Kauffmann et al. 1993):

[EQUATION]

Here µ is the mean molecular weight, [FORMULA] is the fraction of the energy [FORMULA] delivered by supernovae transmitted to the gas ([FORMULA] erg) while [FORMULA] is the number of supernovae per solar mass of stars formed (note that in VS we used [FORMULA] K). The factor [FORMULA] in (5) comes from the dependance of the efficiency of star formation on the properties of the host galaxy. Thus, small galaxies with a shallow potential well ([FORMULA]) are strongly influenced by supernovae and stellar winds which eject part of the gas so that a small fraction of the baryonic matter is converted into stars ([FORMULA]). On the other hand, at low z large halos ([FORMULA]) which formed at a high redshift ([FORMULA]) have already converted most of their gas into stars ([FORMULA]). Indeed, their high density (due to their large redshift of formation) translates into a small dynamical time, hence to very efficient star formation within our model, which leads to the factor [FORMULA] in (6). As in the model used in Valageas & Schaeffer (1999a) and VS we assume that the gas ejected from the inner parts of small galaxies ([FORMULA]) remains bound to (or close to) the galactic halo so that it can cool and fall back into the galaxy. This leads to a self-regulated star formation process (within each individual galaxy) which takes care of the "overcooling problem". Note that an alternative, as suggested in Blanchard et al.(1992) (also Prunet & Blanchard 1999), would be that this gas gets mixed with the IGM and leads to a progressive heating of the IGM which prevents most of the gas to cool and form galaxies. This will also correspond to our supernova heating scenario (SN), see Sect. 2.5.

2.3. Quasars

In a fashion similar to Efstathiou & Rees (1988) and Nusser & Silk (1993) we derive the quasar luminosity function from the multiplicity function of galactic halos. Thus, we assume that the quasar mass [FORMULA] is proportional to the mass of gas [FORMULA] available in the inner parts of the galaxy: [FORMULA]. In our case this also implies that [FORMULA] where [FORMULA] is the stellar mass. We use [FORMULA] which is consistent with observations (Magorrian et al. 1998 find that [FORMULA]). We also assume that quasars shine at the Eddington limit so that their life-time is given by [FORMULA] yr where [FORMULA] is the quasar radiative efficiency and that a fraction [FORMULA] of galactic halos actually host a quasar. Thus, the luminosity of a quasar of mass [FORMULA] is:

[EQUATION]

and the quasar multiplicity function [FORMULA] is obtained from the galaxy mass function [FORMULA] by:

[EQUATION]

Here [FORMULA] is the evolution time-scale of galactic halos of mass M defined by:

[EQUATION]

Since the quasar life-time [FORMULA] yr is quite short, we have [FORMULA]. This also means that [FORMULA]. The factor [FORMULA] shows that the quasar luminosity function is biased towards large redshifts as compared with the galaxy luminosity function. In particular, it peaks at [FORMULA] and shows a significant drop at smaller redshift while the galaxy luminosity function keeps increasing until [FORMULA] and the star formation rate only decreases after [FORMULA]. As we shall see in Sect. 3 this implies that quasar heating of the IGM occurs earlier than supernova heating. Note that we only have two parameters: [FORMULA] and [FORMULA]. Hence a larger fraction of quasars [FORMULA] with a smaller life-time [FORMULA] would give the same results. Moreover, the assumption that quasars shine at the Eddington limit gives the ratio [FORMULA] while the parameter F is constrained by the observed ratio (quasar mass)/(stellar mass). The normalization factor [FORMULA] is constrained by the observed quasar luminosity function. Our results agree reasonably well with available B-band observations for [FORMULA] (VS and Sect. 4.2).

2.4. Lyman-[FORMULA] clouds

We also include in our model a description of Lyman-[FORMULA] clouds. These correspond to density fluctuations in the IGM as well as to virialized halos which may or may not have cooled. More precisely, we consider three different classes of objects (Valageas et al. 1999a).

Low-density mass condensations with a small virial temperature see their baryonic density fluctuations erased over a scale [FORMULA] as the gas is heated by the UV background radiation (or other processes) to a temperature [FORMULA]. More precisely, we define the scale [FORMULA] by:

[EQUATION]

where [FORMULA] is the sound speed, [FORMULA] the IGM temperature, [FORMULA] the age of the universe, [FORMULA] the proton mass and [FORMULA]. These mass condensations form a first population of objects, defined by the scale [FORMULA], which can be identified with the Lyman-[FORMULA] forest at low z. We set the characteristic temperature [FORMULA] by:

[EQUATION]

where [FORMULA] is the reionization redshift and [FORMULA] is the temperature of the IGM. At low z the term [FORMULA] K models photoionization heating for the clouds (the IGM is also heated by the UV flux but in addition it undergoes adiabatic cooling because of the expansion of the universe, so that at low redshift [FORMULA] we can have [FORMULA] K). As explained in Valageas et al.(1999a), these absorbers are not necessarily spherical clouds of radius [FORMULA]. Some may be long filaments with a length [FORMULA] and a thickness [FORMULA]. Moreover, they can also be interpreted as density fluctuations within the IGM rather than distinct entities. Next, potential wells with a larger virial temperature [FORMULA] do not see their baryonic density profile smoothed out. Thus, they define a second class of absorbers which for [FORMULA] correspond to the galactic halos (objects with [FORMULA], if such a range exists, simply are virialized objects which have not cooled hence have not formed stars). Note that one such object can produce a broad range of observed column densities depending on the impact parameter of the line of sight. This population corresponds to Lyman-limit systems. Finally, the deep cores of these halos are neutral because of self-shielding and they form our third class of absorbers, corresponding to damped systems.

2.5. Evolution of the IGM

2.5.1. Temperature evolution

As in VS the gas in the IGM is heated by the background radiation while it cools because of the expansion of the universe and several atomic processes (collisional excitation, ionization, recombination, molecular hydrogen cooling, bremsstrahlung and Compton cooling or heating). Meanwhile, hydrogen and helium are reionized by the UV flux. In our calculation, we take into account the opacity due to the gas present in the underdense regions which fill most of the volume as well as the absorption due to discrete clouds (the "Lyman-[FORMULA] clouds" described above). We also follow the evolution of the HI, HeII and HeIII filling factors describing the ionized bubbles around galaxies and quasars, as well as the clumping of the gas (which also enters explicitly into the model for Lyman-[FORMULA] clouds). The evolution of the background radiation field [FORMULA] is obtained from the radiation emitted by stars and quasars, which we described in the previous sections, see VS for details. We write the evolution of the temperature of the IGM as:

[EQUATION]

where [FORMULA] is the scale factor (which enters the term describing adiabatic cooling due to the expansion). The heating time-scale [FORMULA] which corresponds to photoionization heating is given by:

[EQUATION]

where [FORMULA] (HI,HeI,HeII), [FORMULA] is the ionization threshold of the corresponding species, [FORMULA] its number density in the IGM and [FORMULA] the baryon number density. The cooling time-scale [FORMULA] describes collisional excitation, collisional ionization, recombination, molecular hydrogen cooling, bremsstrahlung and Compton cooling or heating. We compute the redshift evolution of the ionization state of hydrogen and helium and we use the cooling rates from Anninos et al.(1997). In particular, as shown in the upper panel of Fig. 4 in VS at low z after reionization the main cooling processes in the IGM are adiabatic and Compton cooling. Indeed, when the medium is reionized collisional excitation cooling is strongly suppressed (see discussion in VS and Efstathiou 1992).

Finally, we added to the evolution equation we used in VS a new term [FORMULA]. This corresponds to an additional source of energy, which we assume here to be uniform. In particular, this term models in our framework the energy output provided by supernovae or quasars, which has been advocated in the litterature in order to raise the entropy level of the IGM (e.g. Ponman et al. 1998; Tozzi & Norman 1999). As explained in Sect. 2.2, in our original model the influence of supernovae was restricted to their parent galaxy (note that this is consistent with numerical simulations by Mac Low & Ferrara 1999 which suggest that gas ejection is negligible for galactic halos with [FORMULA]). In contrast, one model we investigate in this article corresponds to a "maximally efficient" scenario where the energy produced by supernovae reheats the IGM as a whole. In the actual universe, the effect of supernovae is likely to lie somewhere in-between these two cases, but these two models allow us to get an estimate of the allowed range for the reheating process (see also Tegmark et al. 1993 for a study of reheating and reionization of the IGM by supernovae-driven winds).

We note that using a uniform source of energy (i.e. we do not let the energy source term vary in space as a function of the distance to the nearest galaxy or quasar, although we model this effect for photoionization heating) is probably a better approximation than it may seem at first sight. Indeed, at late times [FORMULA] when this process dominates most of the matter is embedded within positive density fluctuations (filaments, virialized halos, see VS) which show a strong clustering pattern as seen in numerical simulations (e.g. Bond et al. 1996). Note that this is included in our model of the density field, described in Sect. 2.1. For instance, we obtained in Valageas et al.(1999a) the amplitude of the two-point correlation function of Lyman-[FORMULA] clouds and we describe in Valageas et al.(1999c) the bias of the various objects we observe in the universe (Lyman-[FORMULA] clouds, galaxies, quasars, clusters). Thus, most of the volume consists of low-density regions while most of the matter is embedded within small or thin structures (filaments, halos) which are located close to galaxies since most clusters and galaxies form on density peaks within these mass condensations, though there may also be some isolated galaxies amid low-density regions (note that this "bias" translates into the correlations of these objects). As a consequence, the energy provided by supernovae or quasars does not need to travel very far in order to heat most of the matter. Indeed, for this it is sufficient to "spread" the energy over filaments while leaving cool voids in between. In this case, the temperature [FORMULA] would rather correspond to a "mass-averaged" temperature, describing the network of halos and filaments which contain most of the matter while voids would be cooler. Of course, one may also expect voids to be easily heated to the temperature of the filaments since due to their low density and small mass they only require a small amount of energy in order to reach the temperature of the neighbouring regions. Thus, the assumption of a uniform energy source appears to be a reasonable first order approximation. However, it is clear that a carefull study of this problem would be interesting, but this would probably require very detailed numerical simulations which are beyond the scope of this study.

In this article, we consider the additional energy described by the term [FORMULA] in (13) to be provided by supernovae or quasars. Thus, we can write:

[EQUATION]

which explicitly shows these two possible sources of energy. Using our model for galaxies which we described in Sect. 2.2, we can write the source term [FORMULA] due to supernovae as:

[EQUATION]

where [FORMULA] is the mean baryonic density of the universe (the fraction of matter within stars is always negligible) and [FORMULA] is the efficiency factor, similar to [FORMULA] in (7), which measures the fraction of the energy produced by supernovae which is available to heat the gas. Thus, we have [FORMULA]. Next, from the model of quasars presented in Sect. 2.3 we have for the quasar contribution:

[EQUATION]

where [FORMULA] is the efficiency factor similar to [FORMULA] in (8). However, if there are some additional energy sources (e.g. the decay of some exotic particles) we could have an effective [FORMULA] larger than unity. Of course, in this case the time-dependence of this hypothetic energy source is unlikely to be proportional to the star or quasar formation rate and one should explicitly detail the origin of this process to get its time-evolution. In this article, we shall restrict ourselves to the formulation (15) which models the possible effect of supernovae or quasars on the IGM, but one cannot disregard the fact that our source term [FORMULA] may in fact correspond to some new process. From the expressions (16) and (17) we can directly obtain a simple estimate of the magnitude of these effects. Indeed, from (16) we see that supernovae heat the IGM to a temperature [FORMULA] of the order:

[EQUATION]

where we used (7) and the fact that at late times [FORMULA] the fraction of baryonic matter which has been converted into stars is [FORMULA]. Thus, supernovae can reheat the IGM up to [FORMULA] K at most . On the other hand, from (17), (8) and (9) we see that quasars heat the IGM up to [FORMULA] of the order:

[EQUATION]

where we used the parameters introduced in Sect. 2.3 and we defined:

[EQUATION]

The factor [FORMULA] comes from the fact that [FORMULA], which agrees with observations. Thus, quasars can potentially heat the IGM to a very high temperature , much larger than the temperature induced by supernova heating, because quasars are very efficient engines to convert the rest mass energy of matter into radiation or energy while a small fraction of the matter converted into stars leads to supernovae ([FORMULA]) which themselves have a small efficiency factor ([FORMULA]) so that [FORMULA]. Note that the estimates (18) and (19) are very robust, independently of the details of the model, since they are directly constrained by the observed galaxy and quasar luminosity functions. On the other hand, the new parameters [FORMULA] and [FORMULA] are only constrained to be smaller than unity. One would need a detailed study of many physical processes which are still poorly known to set a precise value for these efficiency factors. In this article, we shall treat them as free parameters, which we take to be constant in time. Thus, our goal is to evaluate the possible effects of these energy sources, which in turn will give us some constraints on their magnitude.

2.5.2. Entropy evolution

From the model of the IGM described in the previous sections, we can also obtain the evolution of the entropy of the gas. The entropy of a Maxwell-Boltzmann gas is given by the Sackur-Tetrode equation:

[EQUATION]

where N is the number of particles, within the volume V, and:

[EQUATION]

Thus we define the specific entropy S as:

[EQUATION]

where [FORMULA] is the baryonic number density (and we note [FORMULA] the decimal logarithm). As explained in details in VS, we consider that at late times most of the volume of the IGM consists of large underdense regions with a density contrast [FORMULA] ("u" for underdense) given by:

[EQUATION]

This simply states that at high z (when [FORMULA]) we have [FORMULA] (i.e. the universe is almost exactly a uniform medium on scale [FORMULA]) while at low z we have [FORMULA] since most of the matter is now within overdense objects (clusters, filaments, etc.) while most of the volume is formed by underdense regions. The scale [FORMULA] was defined in (11) while the exponent [FORMULA] is given by the power-law behaviour of the scaling function [FORMULA] at small x. In addition to these "voids" and the virialized halos which are identified to galaxies or clusters, there are also density fluctuations (clouds, filaments) which form the Lyman-[FORMULA] forest or small virialized halos which have not cooled ([FORMULA]) and can have a larger temperature than the underdense regions (the temperature of the gas within these small halos is of the order of the virial temperature of the potential well). It is important to take into account these density fluctuations since at late times ([FORMULA]) the "overdensity" [FORMULA] is as low as [FORMULA] while the density contrast of forest clouds reaches [FORMULA]. This wide variation of the local density means that the average entropy of the IGM can be significantly different from the entropy which would be computed from (23) within "voids". Thus, we first define the entropy [FORMULA] characteristic of the underdense regions which fill most of the volume by:

[EQUATION]

where [FORMULA] is the baryonic density obtained from (24). Then, we define a "mean IGM entropy" [FORMULA] by:

[EQUATION]

where the mean density contrast [FORMULA] is given by:

[EQUATION]

It corresponds to the mean density (total mass over the volume) of the matter which is not embedded within virialized halos which have cooled, from very underdense regions up to forest clouds seen as density fluctuations in the IGM. Since the entropy is an additive quantity, the relevant quantity is indeed the mean [FORMULA] which describes the average entropy of the IGM, see (21). In particular, at late times the quantity [FORMULA] which corresponds to the small fraction of matter located within voids is much larger. We also define the mass-averaged overdensity [FORMULA] characteristic of the matter outside galaxies and clusters by:

[EQUATION]

The quantity [FORMULA] corresponds to the average overdensity of IGM particles, weighted by the number of particles and not by the volume they occupy. Finally we introduce the mean temperature [FORMULA]:

[EQUATION]

This takes into account the fact that some of the gas outside galaxies and clusters is located in Lyman-[FORMULA] forest clouds with [FORMULA] K (due to photoionization heating) which may be larger than [FORMULA] (which also involves adiabatic cooling), and possibly within some virialized halos with [FORMULA] which have not cooled (due to their low virial temperature which implies inefficient cooling).

2.5.3. Effect of the IGM entropy on galaxy formation

As gravitational clustering builds increasingly large structures, the baryonic matter content of the universe gradually becomes embedded into virialized halos where it cools and forms stars, as described in Sect. 2.2 where we detailed our model for galaxy formation. In particular, the "cooling temperature" [FORMULA] which characterizes the smallest virialized halos which can cool was given by the condition [FORMULA] where the cooling time which depends on the density and the temperature of the gas satisfies:

[EQUATION]

where [FORMULA] is the cooling function. Thus cooling is more efficient for larger baryonic densities [FORMULA] (because collisions are more frequent). In the original model the cooling time attached to a given halo was computed using the virial temperature [FORMULA] for T and a density contrast [FORMULA] to obtain the gas density. However, if the IGM is preheated and gets a large entropy at earlier times, the gas may not follow the dark matter to form a mass condensation with mean baryonic density [FORMULA]. Indeed, during the adiabatic collapse of the gas (before it cools) its temperature increases as [FORMULA]. Hence the compression will stop if [FORMULA] reaches [FORMULA] before the density contrast reaches [FORMULA]. Indeed, gas with [FORMULA] does not fall into the potential well. Thus, we obtain an upper bound [FORMULA] for the baryonic density reached within a virialized halo of temperature [FORMULA], defined by:

[EQUATION]

We can also write the density contrast [FORMULA] given by (31) as:

[EQUATION]

Thus, in order to compute the cooling time [FORMULA] from (30) we use the overdensity [FORMULA] given by:

[EQUATION]

Using this prescription we compute the "cooling temperature" [FORMULA]. It is obvious from (30) and (33) that the effect of the entropy of the intergalactic gas is to make cooling less efficient, which leads to a possible increase of the characteristic temperature [FORMULA] . In particular, at late times ([FORMULA]) if the entropy production is sufficiently large it may happen that the cooling temperature [FORMULA] does not exist any more. Indeed, as we explained above the large entropy of the gas diminishes the gas density which enters (30). Moreover, the ratio [FORMULA] displays a minimum at a finite value [FORMULA] K since at large temperatures the cooling function behaves as [FORMULA] (bremsstrahlung is the main cooling process) while below [FORMULA] K it nearly goes to 0 (note that in our original model the high temperature halos which cannot cool are identified to clusters while the low temperature objects are Lyman-[FORMULA] absorbers). As a consequence, if the entropy level is such that the gas density within the halos with a virial temperature [FORMULA] K is too small to allow efficient cooling, no halo can cool. Of course, this does not mean that there are no galaxies! It simply means that all just-virialized halos (i.e. overdensities defined by the density threshold [FORMULA]) are identified to "clusters" (or "groups") in the sense that they consist of one or several smaller higher-density subunits (galaxies), which could cool at earlier times when they formed, embedded within a larger structure containing some hot gas.

Let us note [FORMULA] the largest redshift where no halo can cool (i.e. [FORMULA] as defined above does not exist). We shall have [FORMULA] since the entropy production is linked to galaxy or quasar formation and cooling is less efficient at low redshift where the baryonic density is lower. Moreover, since the mean entropy [FORMULA] increases with time, at smaller redshifts [FORMULA] we have the same situation. Then, along the lines developed in Valageas & Schaeffer (1999a) to distinguish galaxies from clusters at [FORMULA], we assume that gravitational clustering is stable and that, to a first order approximation, galaxies which form at [FORMULA] do not evolve significantly at later times when they get embedded within larger structures which cannot cool as a whole. Thus, after [FORMULA] these galaxies may get closer to form a group but we neglect their possible mergings. Note that the gas which cooled before [FORMULA] and fell into these potential wells to build these galaxies formed "small" dense entities (the baryonic distribution extends to smaller radii than the underlying dark matter halo) which are likely to keep their identity for a longer time than their surrounding dark matter halos which may join to make a larger object. Furthermore, we note that the presence of substructures within dark matter halos themselves and the dependance of the characteristic density of a halo on its mass (it is proportional to the average density of the universe at the time this mass-scale turned non-linear, e.g. Navarro et al. 1996) suggest that this picture may also be a good approximation for the dark matter density fluctuations themselves (see discussion in Valageas 1999). Thus, we assume that after [FORMULA] these galaxies keep their mass, radius and density unchanged. As a consequence, their density contrast at a later time is not [FORMULA] but:

[EQUATION]

Thus, at small redshifts [FORMULA] we no longer define galaxies by the virialization condition [FORMULA]. Instead, we use the constraint [FORMULA] as defined in (34). As explained in Sect. 2.1 this can be done in a straightforward fashion within our description of the non-linear density field: we simply use this density contrast [FORMULA] in (2) to obtain the multiplicity function. Of course, as can be checked in (2) and (4) the parameter x attached to such galactic halos does not evolve with time which also implies that the fraction of matter embedded within these objects is constant with time. Thus our prescription is self-consistent. This relies on the fact that in the non-linear regime relevant for galaxies the two-point correlation function grows as [FORMULA] at fixed physical length R, where [FORMULA] is the scale-factor, as predicted by the stable-clustering ansatz (Peebles 1980). This behaviour is indeed consistent with numerical simulations (e.g. Valageas et al. 1999b). Then, at low redshifts we define [FORMULA] as being equal to the value it had at [FORMULA] (more exactly for [FORMULA]), with the constraint that it is larger than [FORMULA]:

[EQUATION]

This is indeed the virial temperature of the smallest galaxies, which cooled at [FORMULA], with [FORMULA]. This latter constraint is due to the fact that the gas within small halos with [FORMULA] will be heated up to [FORMULA] and it will escape from the potential well (but of course there will remain a small galaxy made of old stars). However, this condition does not play any role in practice since [FORMULA] does not increase much after [FORMULA].

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Online publication: October 14, 1999
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