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Astron. Astrophys. 347, 1-20 (1999)

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8. Numerical results: open universe

We can now use the model we described in the previous sections to obtain the reionization history of the universe, as well as many other properties such as the population of quasars, galaxies or Lyman-[FORMULA] clouds, for various cosmologies. We shall first consider the case of an open universe [FORMULA], [FORMULA], with a CDM power-spectrum (Davis et al.1985), normalized to [FORMULA]. We choose a baryonic density parameter [FORMULA] and [FORMULA] km s- 1 Mpc-1. We use the scaling function [FORMULA] obtained by Bouchet et al. (1991) as explained in VS II. Our model is consistent with the studies presented in VS II and Valageas et al. (1999a), so that those papers are part of the same unified model and describe in more details our predictions for galaxies and Lyman-[FORMULA] clouds at [FORMULA].

8.1. Quasar luminosity function

Although our model was already checked in previous studies for galaxies and Lyman-[FORMULA] clouds as explained above, our description for quasars was not compared to observations in great detail (although a first check was performed in VS II). Thus, we first compare in this section our predictions for the quasar luminosity function to observational data, as shown in Fig. 1.

[FIGURE] Fig. 1. The evolution with redshift of the B-band quasar luminosity function in comoving Mpc-3. The data points are from Pei (1995).

We can see that our model is consistent with observations. At low redshifts the number of quasars we predict does not decline as fast as the data, however we get a significant decrease which is already an improvement over the results of Efstathiou & Rees (1988) for instance. We can note that Haehnelt & Rees (1993) managed to obtain a good fit to the observed decline at low z but they had to introduce an ad-hoc redshift and circular velocity dependence for the black hole formation efficiency. Since our model appears to work reasonably well we prefer not to introduce additional parameters. Moreover, as we noticed earlier our ratio (black hole mass)/(stellar mass) is consistent with observations ([FORMULA]) while the quasar life-time we use [FORMULA] yrs agrees with theoretical expectations. The high-luminosity cutoff, which appears at [FORMULA], comes from the fact that in our model very massive and bright galaxies have consumed most of their gas. Thus, the maximum quasar luminosity starts decreasing with time at low z because of fuel exhaustion. We note that Haiman & Loeb (1998) obtained similar results at [FORMULA] although they used a very small time-scale [FORMULA] yrs (in our case this problem is partly solved by the introduction of the parameter [FORMULA] which states that only a small fraction of galaxies actually host a black hole). However, they note that the number density of bright quasars they get increases until [FORMULA].

In a recent paper, Haiman et al. (1998) point out that the lack of quasar detection down to magnitude [FORMULA] in the HDF strongly constrains the models of quasar formation, which tend to predict more than 4 objects (which is still marginally consistent). In particular, they find that one needs to introduce a lower cutoff for the possible mass of quasars (shallow potential wells with a circular velocity lower than 50 km s-1 are not allowed to form back holes) or a mass-dependent black-hole formation efficiency. We show in Fig. 2 the predictions of our model.

[FIGURE] Fig. 2. The quasar cumulative V-band number counts. The dashed line shows the counts of quasars with magnitude brighter than V located at the redshifts [FORMULA] while the solid line corresponds to [FORMULA].

The solid line shows the number [FORMULA] of quasars with a magnitude lower than V located at a redshift larger than 3.5 up to the reionization redshift [FORMULA]. The large opacity beyond this redshift prevents detection of higher z objects. We also take into account the opacity due to Lyman-[FORMULA] clouds at lower z. We can see that our model is marginally consistent with the constraints from the HDF since it predicts 4 detections up to [FORMULA]. We note that our model automatically includes photoionization feedback (threshold [FORMULA]) and a virial temperature dependence in the relation (black hole mass) - (dark matter halo mass). However, the "cooling temperature" [FORMULA] is too low to have a significant effect on the number counts. Of course, we see that at bright magnitudes most of the counts come from low-redshift quasars ([FORMULA]). Thus, the QSO number counts strongly constrain our model since in order to obtain a reionization history consistent with observations (namely the HI and HeII Gunn-Peterson tests and the low-redshift amplitude of the UV background radiation field) we need a relatively large quasar multiplicity function. However, one might weaken these constraints by using an ad-hoc QSO luminosity function with many faint objects ([FORMULA]).

8.2. Reheating of the IGM

As we explained previously the radiation emitted by galaxies and quasars will reheat and reionize the universe, following (28) and (30). We start our calculations at [FORMULA] with the initial conditions used by Abel et al. (1998), see also Peebles (1993). In particular:




and we use a helium mass fraction [FORMULA]. We present in Fig. 3 the redshift evolution of the IGM temperature [FORMULA], as well as the temperature [FORMULA] which defines the smallest virialized objects which can cool at redshift z, see (5).

[FIGURE] Fig. 3. The redshift evolution of the IGM temperature [FORMULA] (solid curve). We also show the virial temperature [FORMULA] of the smallest virialized halos which can cool at redshift z (dashed curve) while [FORMULA] is a mass-averaged temperature (dot-dashed curve).

At high z the IGM temperature decreases with time due to the adiabatic expansion of the universe. Next, for [FORMULA] ([FORMULA]) the medium starts being slowly reheated by the radiative output of stars and quasars until it reaches at [FORMULA] a maximum temperature [FORMULA] K where collisional excitation cooling is so efficient that the IGM temperature cannot increase significantly any more. As we shall see later, this phase occurs before the medium is reionized, as was also noticed by Gnedin & Ostriker (1997) using a numerical simulation. There is a small increase at [FORMULA] ([FORMULA]) when the universe is fully reionized and the UV background radiation shows a sharp rise. However, because cooling is very efficient, the dramatic increase in [FORMULA] only leads to a small change in [FORMULA]. Eventually at low redshifts the temperature starts decreasing again due to the expansion of the universe as the heating time-scale becomes larger than the Hubble time [FORMULA]. The temperature [FORMULA] which defines the smallest objects which can cool at redshift z increases with time because the decline of the number density of the various species, due to the expansion of the universe, makes cooling less and less efficient. Indeed, the cooling rate (in erg cm-3 s-1) associated with a given process involving the species i and j can usually be written as [FORMULA], which leads to a cooling time-scale:


where we have neglected the temperature dependence. Thus, the ratio (cooling time)/(Hubble time) increases as time goes on (at fixed T and abundance fractions). Since halos with virial temperature [FORMULA] must satisfy [FORMULA], see (5), [FORMULA] has to get higher with time to increase the rate [FORMULA] (which usually contains factors of the form [FORMULA]). The sudden increase of [FORMULA] at [FORMULA] ([FORMULA]) is due to the decline of the fraction of molecular hydrogen which starts being destroyed by the radiation emitted by stars and quasars. As a consequence the main cooling process becomes collisional excitation cooling instead of molecular cooling. Since the former is only active at high temperatures (the coefficient rate [FORMULA] contains a term [FORMULA] instead of [FORMULA] for molecular cooling) the cooling temperature [FORMULA] has to increase up to [FORMULA] K. By definition [FORMULA] is larger than the IGM temperature and usually much higher as can be seen in Fig. 3. However, at [FORMULA] when [FORMULA] K is quite high due to reheating by the background radiation field we have [FORMULA] since the IGM temperature is large enough to allow for efficient cooling. Then all virialized bound objects, with [FORMULA], form baryonic clumps which can cool. The temperature [FORMULA] represents a mass-averaged temperature: the matter within the IGM is associated to [FORMULA] while virialized objects (hence with [FORMULA]) are characterized by a temperature defined as Min[FORMULA]. Since [FORMULA] does not enter any of our calculations used to obtain the redshift evolution of the universe this crude definition is sufficient for our purpose which is merely to illustrate the difference between volume ([FORMULA]) and mass averages. As can be seen from Fig. 3 we always have [FORMULA] as it should be. At large redshifts [FORMULA] since most of the matter is within the uniform IGM component, whereas at low redshifts [FORMULA] ([FORMULA]) the IGM temperature declines as we explained previously while [FORMULA] remains large since most of the matter is now embedded within collapsed objects where shock heating is important (and they do not experience adiabatic cooling due to the expansion).

We show in Fig. 4 the cooling and heating times associated with various processes for the IGM as well as for the smallest halos [FORMULA] which can cool at z.

[FIGURE] Fig. 4. The cooling and heating times associated with the various relevant processes in units of [FORMULA] for the IGM (upper figure ) and the halos defined by [FORMULA] (lower figure ). The labels are as follows: 1) collisional excitation, 2) collisional ionization, 3) recombination, 4) molecular hydrogen, 5) bremsstrahlung, 6) Compton, 7) photoionization heating and 8) cooling due to expansion (only for the IGM, see text).

We can see in the upper panel that for large and small redshifts, [FORMULA] ([FORMULA]) and [FORMULA] ([FORMULA]), all time-scales associated with the IGM are larger than the Hubble-time which means that the IGM temperature declines due to the adiabatic cooling entailed by the expansion of the universe. However, at intermediate redshifts [FORMULA] ([FORMULA]) the smallest time-scale corresponds to heating by the background radiation ([FORMULA]) which means that [FORMULA] increases during this period. Next, at [FORMULA], the IGM temperature becomes large enough to activate collisional excitation cooling so as to reach a temporary equilibrium where [FORMULA] while [FORMULA] remains constant. Then, as we shall see later the universe gets suddenly reionized at [FORMULA] ([FORMULA]). This means that [FORMULA] increases sharply as [FORMULA] declines (as well as [FORMULA] and [FORMULA]) as can be seen from (29). The cooling time due to collisional excitation follows this rise as the medium remains in quasi-equilibrium while the temperature declines slightly (the strong temperature-dependent factors like [FORMULA] in [FORMULA] ensure it immediately adjusts to [FORMULA], moreover these cooling rates are also proportional to [FORMULA], [FORMULA] and [FORMULA]) until both heating and cooling time-scales become larger than the Hubble time. Then this quasi-equilibrium regime stops as the medium merely cools because of the expansion of the universe. These various phases, which appear very clearly in Fig. 4, explain the behaviour of [FORMULA] shown in Fig. 3 which we described earlier. The peak at [FORMULA] ([FORMULA]) of [FORMULA] in the upper panel (curve 6) corresponds to the time when its sign changes (hence [FORMULA]). At higher redshifts [FORMULA] is lower than the CMB temperature (due to adiabatic cooling by the expansion of the universe), so that the gas is heated by the CMB photons while at lower z the IGM temperature is larger than [FORMULA] (due to reheating) so that the gas is cooled by the interaction with the CMB radiation.

The lower panel shows the cooling and heating times associated with the halos [FORMULA]. As we have already explained we can see that at high redshifts the main cooling process is molecular hydrogen cooling. Note however that for the IGM this process is always irrelevant. This difference comes from the fact that the larger density and temperature of these virialized halos allow them to form more molecular hydrogen than is present in the IGM, so that molecular cooling becomes efficient. This was also described in detail in Tegmark et al. (1997) for instance. Of course at these redshifts we have [FORMULA] by definition of [FORMULA]. Then, at [FORMULA] ([FORMULA]) as molecular hydrogen starts being destroyed by the background radiation the main cooling process becomes collisional excitation. Note that the corresponding cooling time gets smaller than [FORMULA] because the medium is also heated by the radiation so that the actual cooling time results from a slight imbalance between cooling and heating processes. The sharp decrease of the various time-scales at [FORMULA] corresponds to a sudden increase of [FORMULA] due to the rise of [FORMULA] (which influences the cooling halos since [FORMULA]) also seen in Fig. 3 and in the upper panel of Fig. 4. Around [FORMULA] ([FORMULA]) we have [FORMULA] and [FORMULA] so that all virialized halos above [FORMULA] can cool ([FORMULA]). The feature at [FORMULA] ([FORMULA]) is due to reionization.

Finally, we present in Fig. 5 the characteristic masses we encounter. The mass [FORMULA] is obtained from (22):


while [FORMULA] corresponds to the halos which can cool at redshift z, as we have already explained. The mass [FORMULA] which follows closely the behaviour of [FORMULA] describes the smallest virialized objects. It differs from [FORMULA] because the density contrast is now [FORMULA] instead of [FORMULA]. The mass [FORMULA] corresponds to the first non-linear scale defined by [FORMULA]. Note that after reionization [FORMULA] while the usual Jeans mass would be [FORMULA]. This is mainly due to the low density [FORMULA] of the IGM, see (46) and Fig. 13 below.

[FIGURE] Fig. 5. The redshift evolution of the characteristic masses [FORMULA], [FORMULA], [FORMULA] and [FORMULA] in [FORMULA].

By definition we have [FORMULA]. At [FORMULA] we have [FORMULA] since [FORMULA] as we noticed earlier in Fig. 3. We also note that at large z our calculation is not entirely correct since our multiplicity functions are valid in the non-linear regime, for masses [FORMULA]. However, at these early times the universe is nearly exactly uniform (by definition!) so that this is not a very serious problem. We can see that the first cooled objects which form in significant numbers are halos of dark-matter mass [FORMULA] which appear at [FORMULA] ([FORMULA]), when [FORMULA] becomes smaller than [FORMULA]. However, they only influence the IGM after [FORMULA] ([FORMULA]) when reheating begins.

8.3. Reionization of the IGM

After the radiation emitted by quasars and stars reheats the universe, as described in the previous section, it will eventually reionize the IGM. We present in Fig. 6 the evolution with redshift of the background radiation field and of the comoving stellar formation rate. Within the framework of our model the latter is a good measure of the radiative output from galaxies, see (12), as well as from quasars, see (17), since we note that the quasar mass happens to be roughly proportional to the stellar mass.

[FIGURE] Fig. 6. The redshift evolution of the UV flux [FORMULA] (upper panel ) and of the comoving star formation rate [FORMULA] (lower panel ). The data points are from Giallongo et al. (1996) (square), Cooke et al. (1997) (filled square), Vogel et al. (1995) (triangle, upper limit), Donahue et al. (1995) (filled triangle, upper limit) and Kulkarni & Fall (1993) (circle). The dashed line in the lower panel shows the effect of the absorption of high energy photons by the neutral hydrogen present in the IGM and in Lyman-[FORMULA] clouds.

The upper panel of Fig. 6 shows the UV flux [FORMULA] in units of [FORMULA] erg cm- 2 s-1 Hz-1 sr-1 defined by (38). We can see that the UV flux rises very sharply at [FORMULA] ([FORMULA]) which corresponds to the reionization redshift [FORMULA] when the universe suddenly becomes optically thin, so that the radiation emitted by stars and quasars at large frequencies is no longer absorbed and contributes directly to [FORMULA]. This appears clearly from the lower panel. Here the solid line shows the comoving star formation rate, obtained from (10), while the dashed line shows the same quantity multiplied by a luminosity-weighted opacity factor [FORMULA] which describes the opacity due to the IGM and Lyman-[FORMULA] clouds (see below (51), (52) and Fig. 10). Thus, we can see that while the star formation rate evolves rather slowly with z the absorption term varies sharply around [FORMULA]. Hence the universe is suddenly reionized at [FORMULA] (when the ionized bubbles overlap: [FORMULA]) on a time-scale very short as compared to the Hubble time because of the strongly non-linear effect of the opacity. We note that for [FORMULA] our star-formation model is somewhat simplified, as we explained earlier, because we defined all galaxies by a constant density contrast [FORMULA] while cooling constraints should be taken into account, as in VS II, and we also used approximations for the stellar content of galaxies which are not strictly valid for all galactic halos at these low redshifts. The reader is referred to VS II for a more precise description of the low z behaviour. However, our present treatment is sufficient for our purposes and still provides a reasonable approximation at [FORMULA].

We display in Fig. 7 the background radiation spectrum [FORMULA] at four redshifts: [FORMULA] (before reionization), [FORMULA] (after reionization), [FORMULA] and [FORMULA].

[FIGURE] Fig. 7. The background radiation spectrum [FORMULA] (in units of [FORMULA] erg cm- 2 s-1 Hz-1 sr-1) at the redshifts [FORMULA] (solid line, prior to reionization), [FORMULA] (lower dashed line, after reionization), [FORMULA] (upper dashed line) and [FORMULA] (solid line).

We see that the ionization edges corresponding to HI, HeI and HeII can be clearly seen at high z before the universe is reionized. Then the background radiation is very strongly suppressed for [FORMULA] eV due to HI and HeI absorption. Of course at very large frequencies [FORMULA] keV where the cross-section gets small we recover the slope [FORMULA] of the radiation emitted by quasars. At low redshifts [FORMULA] after reionization the drop corresponding to HeI disappears as HeI is fully ionized and its number density gets extremely small, as we shall see below in Fig. 9. However, even at [FORMULA] the ionization edges due to HI and HeII are clearly apparent and [FORMULA] is still significantly different from a simple power-law. At low redshifts [FORMULA] the background radiation is much smoother since its main contribution comes from radiation emitted while the universe was ionized and optically thin. However, its intensity is smaller than at [FORMULA] because the quasar luminosity function drops at low z, see Fig. 1, while the universe keeps expanding.

We show in Fig. 8 the redshift evolution of the ionization and recombination times [FORMULA] and [FORMULA] of the IGM, divided by [FORMULA].

[FIGURE] Fig. 8. The redshift evolution of the ionization and recombination times [FORMULA] (solid line) and [FORMULA] (dashed line) of the IGM, divided by the Hubble time [FORMULA]. The horizontal solid line only shows [FORMULA] for reference. The recombination times [FORMULA] (uniform medium) and [FORMULA] (ionized bubbles) are defined in the main text.

More precisely, the ionization time [FORMULA] is defined by:


while the recombination time within the IGM is:


where [FORMULA] is the recombination rate, [FORMULA] the clumping factor and [FORMULA] the mean electron number density, from (27) and (26). We also display for reference the recombination time which would correspond to a uniform medium with the mean density of the universe:


Finally, we show the recombination time within ionized bubbles (where all hydrogen atoms are ionized):


The recombination time grows with time at high z as the mean density decreases with the expansion, although this is somewhat balanced by the increase of the clumping factor [FORMULA] (see below Fig. 13). In particular, the decrease of [FORMULA] around [FORMULA] ([FORMULA]) is due to the growth of [FORMULA]. The sharp drop at [FORMULA] ([FORMULA]) is due to reionization which suddenly increases the number density of free electrons. After reionization the recombination time characteristic of the IGM keeps decreasing (while [FORMULA] increases slightly since the density declines) because of the growth of the clumping factor [FORMULA], see Fig. 13, which overides the decline of the mean universe density. The recombination time within ionized bubbles [FORMULA] follows the change of the mean universe density and of the clumping factor [FORMULA]. At large z it is much smaller than the mean IGM recombin me since the IGM is close to neutral. At low z it becomes larger than [FORMULA] since the IGM is suddenly reionized with a temperature [FORMULA] which declines after reheating and gets lower than [FORMULA] K, see Fig. 3. The ionization time is very large at high z since the UV background radiation is small. Then it decreases very sharply at [FORMULA] when the universe is reionized and the background radiation suddenly grows as the medium becomes optically thin, as seen in Fig. 6. The reionization redshift corresponds to the time when [FORMULA] becomes smaller than [FORMULA], somewhat after it gets smaller than [FORMULA]. Thus [FORMULA] does not play a decisive role since it is never the smallest time-scale around reionization.

Finally, we show in Fig. 9 how the chemistry of the IGM evolves with time as the temperature [FORMULA] and the UV flux [FORMULA] vary with z.

[FIGURE] Fig. 9. The redshift evolution of the chemistry of the IGM. The upper panel shows the ionization state of hydrogen, as well as the fraction of molecular hydrogen and electrons. The lower panel presents the ionization of helium.

We see very clearly in the upper panel the redshift of reionization [FORMULA] ([FORMULA]) when the fraction of neutral hydrogen [FORMULA] declines very sharply while the UV flux [FORMULA] suddenly rises, as was shown in Fig. 6. We note that at low redshifts [FORMULA] the neutral hydrogen fraction decreases more slowly down to [FORMULA]. The fractions of electrons and ionized hydrogen start increasing earlier at [FORMULA] ([FORMULA]) but of course they remain small until [FORMULA]. The abundance of molecular hydrogen decreases sharply at a rather high redshift [FORMULA] ([FORMULA]) due to the background radiation, as we noticed on Fig. 4. The lower panel shows that helium gets fully ionized simultaneously with hydrogen. In particular, although there remains a small fraction of HeII ([FORMULA]) the abundance of HeI gets extremely small. We note that at low redshifts [FORMULA] the fraction of HeII does not evolve much (and even slightly increases) while the HI abundance keeps declining. This is due to the fact that the radiation relevant for helium ionization comes from quasars whose luminosity function drops at low z as shown in Fig. 1 while an important contribution to the hydrogen ionizing radiation is provided by stars and the galaxy luminosity function declines more slowly with time at low z, as seen in Fig. 6 or in VS II. We shall come back to this point in Sect. 8.7.

8.4. Opacities

As we explained previously the radiation emitted by stars and quasars at high frequencies ([FORMULA] eV) is absorbed by the IGM and discrete clouds as it propagates into the IGM. This leads to the extinction factors [FORMULA] and [FORMULA] in the evaluation of the source terms (13) and (21) for the background radiation. We define here the "luminosity averages" [FORMULA] for both continuous and discrete components by:


where [FORMULA] is the relevant opacity (from the IGM or clouds for sources x, see (44)) at the frequency 20 eV below the HeI ionization threshold without taking into account the factors [FORMULA] which describe ionized bubbles. The subscript s refers to the fact that we consider here the opacity which enters into the calculation of the stellar radiative output. The quasar-related opacity mainly differs through the factor [FORMULA], see (42). The weight [FORMULA] corresponds to a luminosity weight as we noticed earlier, see (12), (17) and (6).

We show in the upper panel of Fig. 10 the redshift evolution of the opacity from the IGM ([FORMULA], dashed line) and from "Lyman-[FORMULA] clouds" ([FORMULA], solid line). We can see that both contributions have roughly the same magnitude before reionization, except at very high redshifts [FORMULA] ([FORMULA]) when very few structures exist as shown in Fig. 12. Prior to reionization the opacity is large and the background radiation quite small, as seen in Fig. 6. At [FORMULA] the opacity suddenly declines while [FORMULA] rises sharply, due to the strong non-linear coupling between [FORMULA] and [FORMULA], as we explained in the lower panel of Fig. 6 where we presented the influence of the total opacity [FORMULA]:


[FIGURE] Fig. 10. Upper panel: the redshift evolution of the opacity from the IGM (dashed line) and "Lyman-[FORMULA] clouds" (solid line) which enters the absorption factors in the calculation of the radiative output from stars and quasars. Lower panel: evolution of the filling factors [FORMULA] (ionized bubbles around quasars, solid line), [FORMULA] (around galaxies, upper dashed line) and [FORMULA] (lower dashed line).

At low redshifts [FORMULA] when the universe is reionized the opacity due to discrete clouds becomes much larger than the IGM contribution (although it is very small) because the density of neutral hydrogen is now proportional to the square of the baryonic density (in photoionized regions) and most of the matter is embedded within collapsed objects.

The opacities [FORMULA] were shown in the upper panel of Fig. 10 without the filling factors [FORMULA] which enter the actual evaluation of the source terms (13) and (21), see (40). The filling factors [FORMULA], describing the volume fraction occupied by ionized bubbles, are shown in the lower panel of Fig. 10. We can see that [FORMULA] and [FORMULA] increase with time as structures form and emit radiation while the IGM density declines. When these ionized bubbles overlap ([FORMULA]) the universe is reionized. At low redshifts the coefficient [FORMULA] declines because the background radiation is large while the quasar number density drops (indeed [FORMULA] measures the volume occupied by the "spheres of influence" of quasars).

Next, we can evaluate the mean opacities [FORMULA] and [FORMULA] seen on a random line of sight from [FORMULA] to a quasar located at redshift z. We present in Fig. 11 the contributions from both the uniform IGM component and the discrete Lyman-[FORMULA] clouds.

[FIGURE] Fig. 11. The redshift evolution of the average opacities [FORMULA] and [FORMULA] along a random line of sight produced by "Lyman-[FORMULA] clouds". The dashed lines show the opacities from the uniform IGM. The data points are from Press et al. (1993) (circles), Zuo & Lu (1993) (filled circles) for hydrogen, and from Davidsen et al. (1996) (filled rectangle) and Hogan et al. (1997) (cross) for helium.

At high redshifts prior to reionization the main contribution to the opacity is provided by the IGM which contains most of the matter. However, at low z when the IGM is ionized most of the absorption comes from the Lyman-[FORMULA] clouds. At large z the HeII opacity is very small because most of the helium is in its neutral form HeI. We refer the reader to Valageas et al. (1999a) for a much more detailed description of the properties of the Lyman-[FORMULA] clouds at low z. We can see that our predictions show a reasonable agreement with observations for the hydrogen opacity. At low redshifts [FORMULA] the influence of star-formation which consumes and may eject some of the gas (which we did not take into account here) could explain the relatively high opacity we obtain. The helium opacity we find is also close to observations. This is due to the fact that the UV radiation spectrum shows strong ionization edges, even at relatively low redshifts [FORMULA] ([FORMULA]), see Fig. 7. Hence the ratio [FORMULA] is rather large, see Fig. 9, which explains why we get a better agreement than Zheng et al. (1998) for instance (see also Valageas et al.1999a). In particular, we have [FORMULA] since [FORMULA] while [FORMULA] at low redshift. We note that the observed HeII opacity strongly constrains the quasar contribution to the reionization process since stellar radiation is small at high frequencies (due to the near blackbody behaviour of stellar spectra). In particular, it implies that one needs a population of faint QSOs ([FORMULA]) in order to reionize helium but [FORMULA] should not be too large so that there is still an appreciable density of HeII. In other words, as we noticed above, the UV radiation field must still display strong ionization edges, which means that it has not had enough time to be smoothed out by the radiation emitted since [FORMULA] when the medium is optically thin.

8.5. Stellar properties

Our model also allows us to obtain the fraction of matter within virialized or cooled halos, as well as in stars.

We show in Fig. 12 the fraction of matter within virialized halos ([FORMULA], upper dashed line), cooled objects ([FORMULA], upper solid line) and stars ([FORMULA], lower dashed line). The first two quantities are simply obtained from (3). Of course we have: [FORMULA]. Around [FORMULA] we note that [FORMULA] because as we explained previously at this time all virialized objects (with [FORMULA]) can cool efficiently ([FORMULA]). The fraction of matter within virialized halos increases very fast at high redshifts [FORMULA] ([FORMULA]) as [FORMULA] becomes smaller than [FORMULA], see Fig. 5, when dark matter structures form on scale [FORMULA]. However, until [FORMULA] ([FORMULA]) the mass within cooled halos remains much smaller because cooling is not very efficient so that [FORMULA], see Fig. 3. At low redshifts [FORMULA] ([FORMULA]) both [FORMULA] and [FORMULA] get close to unity since most of the matter is now embedded within collapsed and cooled halos (even though [FORMULA] becomes again much larger than [FORMULA]: we are so far within the non-linear regime that even [FORMULA] is small compared to the characteristic virial temperature of the structures built on scale [FORMULA]). Of course the mass within stars grows with time, closely following [FORMULA]. Note however that it is not strictly proportional to [FORMULA] since an increasingly large fraction of the gas within galaxies is consumed into stars. The fraction of volume [FORMULA] occupied by virialized objects always remains small as it satisfies:


[FIGURE] Fig. 12. The redshift evolution of the fraction of matter enclosed within virialized halos ([FORMULA]), cooled objects ([FORMULA]) and stars ([FORMULA]). The lower solid line is the volume fraction [FORMULA] occupied by virialized objects.

We show in the upper panel of Fig. 13 the clumping factor [FORMULA] defined by (25). The expression (25) shows clearly that at large redshifts where there are very few collapsed baryonic structures [FORMULA] while at low z when most of the baryonic matter is within virialized halos (note this is always true for dark matter on sufficiently small scales) we have [FORMULA]. We can see in the figure that [FORMULA] usually increases with time as the hierarchical clustering process goes on. The temporary decrease at [FORMULA] ([FORMULA]), which also appears in the lower panel and in Fig. 12, is due to the reheating of the universe, shown in Fig. 3, which increases the "damping" length [FORMULA] and mass scale [FORMULA], as seen in Fig. 5. As a consequence, small objects which were previously well-defined entities suddenly see the IGM temperature become larger than their virial temperature. Hence they cannot retain efficiently their gas content and they lose their identity. We note that neglecting the clumping of the gas would lead to a higher reionization redshift [FORMULA] since it would underestimate the efficiency of recombination. The clumping factor [FORMULA] displays a behaviour similar to [FORMULA] but it is usually smaller since it does not include the deep halos which can cool. We display in the lower panel of Fig. 13 the overdensity [FORMULA] characteristic of most of the volume of the universe at redshift z when seen on scale [FORMULA]. While structures form and the matter gets embedded within overdensities which occupy a decreasing fraction of the volume (when seen at this scale [FORMULA]) the "overdensity" [FORMULA] which characterises the medium in-between these objects (halos or filaments) declines. The density contrast [FORMULA] which corresponds to the IGM and shallow potential wells which do not form stars (but constitute Lyman-[FORMULA] clouds) decreases more slowly since it only excludes the high virial temperature halos with [FORMULA].

[FIGURE] Fig. 13. Upper panel: the redshift evolution of the clumping factors [FORMULA] and [FORMULA]. Lower panel: the overdensities [FORMULA] characteristic of most of the volume of the universe at redshift z at scale [FORMULA] and [FORMULA].

We present in Fig. 14 the redshift evolution of the mean metallicities (in units of solar metallicity) [FORMULA] (within the star-forming gas located in the inner parts of galaxies, upper solid line), [FORMULA] (within stars, upper dashed line) and [FORMULA] (within galactic halos, lower dashed line). We use the mass average over the various galactic halos:


where [FORMULA] is the galaxy mass function defined as in (3). The reader is referred to VS II for a detailed description of these metallicities (note that we only consider here the abundance of Oxygen or any other element that is mainly produced by SN II since we did not include SN I in our model). The lower solid line corresponds to a "matter averaged" metallicity [FORMULA] defined by [FORMULA]. Thus, although we do not include explicitly in our model any contamination of the IGM by heavy elements produced within galaxies, [FORMULA] defined in this way provides an upper bound for the mean IGM metallicity (corresponding to very efficient mixing). If galaxies do not eject metals very deeply within the IGM its metallicity could be much smaller. The mean metallicity of Lyman-[FORMULA] clouds associated to galactic halos (limit or damped systems) is [FORMULA]. Our results agree well with observations by Pettini et al. (1997) for damped Lyman-[FORMULA] systems. Note that there is in fact a non-negligible spread in metallicity over the various halos.

[FIGURE] Fig. 14. The redshift evolution of the metallicities [FORMULA] (star-forming gas), [FORMULA] (stars), [FORMULA] (galactic halos) and [FORMULA] (matter average). The data points are from Pettini et al. (1997) for the zinc metallicity of damped Lyman-[FORMULA] systems.

8.6. Consequence for the CMB radiation

After reionization, CMB photons may be scattered by electrons present in the gas. We write the corresponding Thomson opacity up to a redshift z as:


where [FORMULA] is the mean electron number density at redshift z. We take:


which means that we use the same electron fraction in clouds as for the IGM (note that we calculate the IGM electron number density together with the ionization state of hydrogen and helium). Then, CMB anisotropies are damped on angular scales smaller than the angle subtended by the horizon at reionization. We use the analytic fit given by Hu & White (1997) to obtain the damping factor [FORMULA] of the CMB power-spectrum [FORMULA] from the optical depth [FORMULA]. The results are shown in Fig. 15.

[FIGURE] Fig. 15. Upper panel: the optical depth [FORMULA] for electron scattering. Lower panel: damping factor [FORMULA] for the CMB power-spectrum.

We can check that the total opacity [FORMULA] is quite small because reionization occurs rather late at [FORMULA]. This also implies that the damping factor [FORMULA] remains close to unity: [FORMULA] for large l. Another distortion of the CMB radiation is the Sunyaev-Zeldovich effect which transfers photons from the Rayleigh-Jeans part of the spectrum to the Wien tail when they are scattered by hot electrons. The magnitude of this perturbation is conveniently described by the Compton parameter y:


We can first consider the contribution of the IGM gas, using in (57) the temperature and the electronic density of this uniform component. Then, we estimate the distortion due to the hot gas embedded within virialized objects. We can write this latter contribution as:


where we used the same electronic fraction for halos and the IGM. The temperature [FORMULA] is the "mass averaged" temperature of virialized objects while [FORMULA] is the mass fraction within collapsed objects displayed in Fig. 12. The results are presented in Fig. 16.

[FIGURE] Fig. 16. The Compton parameter y up to redshift z describing the Sunyaev-Zeldovich effect from the IGM (dashed line) and virialized halos (solid line).

The Compton parameter [FORMULA] due to the IGM first increases rather fast with z until reionization, together with [FORMULA] and [FORMULA]. After [FORMULA] it reaches a plateau and does not grow any more since at these large redshifts the universe is almost exactly neutral. The contribution [FORMULA] of virialized objects is much larger at low z than [FORMULA] since the temperature of these collapsed halos is much higher than [FORMULA], as shown in Fig. 3. We can note however that [FORMULA] grows much slower and becomes close to its asymptotic value earlier than [FORMULA]. This is due to the fact that the characteristic temperature of virialized halos declines at larger z, see Fig. 3, and the mass fraction they contain also decreases (while the IGM undergoes the opposite trends). A more detailed description of the Sunyaev-Zeldovich effect due to clusters, and its fluctuations (which in fact have the same magnitude as the mean), will be presented in a future article.

8.7. Contributions of quasars and stars

In our model the radiation which reheats and reionizes the universe comes from both quasars and stars. At large frequencies [FORMULA] eV most of the UV flux is emitted by quasars so that stars play no role in the helium ionization. However, at lower frequencies both contributions have roughly the same magnitude. We present in Fig. 17 the redshift evolution of the radiative output due to stars and quasars.

[FIGURE] Fig. 17. Upper panel: the redshift evolution of the "instantaneous" UV fluxes [FORMULA] and [FORMULA] due to stars (solid lines) and quasars (dashed lines). The dotted line is the UV flux [FORMULA] shown in Fig. 6. Lower panel: the "instantaneous" ionization times [FORMULA] due to stars (solid line) and quasars (dashed line). The dotted curve is the recombination time as in Fig. 8 while the horizontal solid line is the Hubble time [FORMULA].

Thus, we define the "instantaneous" radiation fields:


where [FORMULA] is the Hubble time, see (30). From these quantities we define the averages [FORMULA] and [FORMULA] as in (38). This provides a measure of the radiative output above the ionization thresholds [FORMULA] and [FORMULA] due to stars and quasars. We can also derive the ionization times [FORMULA] as in (47). We can see in the upper panel that at reionization [FORMULA] the contributions to HI ionizing radiation from stars and quasars are of the same order. However, since the quasar spectrum is much harder than stellar radiation we have [FORMULA] so that quasars are slightly more efficient at reheating and reionizing the universe (the additional factor [FORMULA] in (29) increases the weight of high energy photons which also remain longer above the threshold [FORMULA] while being redshifted). At low z we can see that the quasar radiative output declines much faster than the stellar source term. Of course, this is due to the sharp drop at low redshifts of the quasar luminosity function. This decrease of [FORMULA] as compared to [FORMULA] comes from two effects: i) as time increases the "creation time-scale" of halos of the relevant masses [FORMULA] (through merging of smaller sub-units, measured by [FORMULA]) grows and ii) there is less gas available to fuel the quasars (which even disappear) while old stars can still provide a non-negligible luminosity source for galaxies. We can note in the upper panel that the redshift evolution of the background radiation field [FORMULA] actually produced by stars and quasars does not exactly follow the "instantaneous" quantities [FORMULA] since one must take into account the expansion of the universe and deviations from equilibrium. This explains the slower increase of [FORMULA] at [FORMULA] as well as the relatively low approximate ionization times [FORMULA] at this epoch.

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© European Southern Observatory (ESO) 1999

Online publication: June 18, 1999