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

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4. Feedback on structure formation

4.1. Star formation

As described in Sect. 2.5.3 the entropy of the IGM inhibits the cooling of the gas which in turns decreases the efficiency of star formation. In particular, after the redshift [FORMULA] the gas which would become embedded within new non-linear structures cannot cool within a few Hubble times at formation because its entropy prevents its density to reach large enough values to start cooling efficiently. Thus, in our model we obtain a population of "old" galaxies which gradually convert their matter content into stars but the overall fraction of gas which can cool does not increase any more. After a while, when a large part of this matter has formed stars, this leads to a decrease of the star formation rate at low z. Note that we neglect cooling flows within groups and clusters which provide an additional source of cool gas which may form stars. However, this feedback effect onto galaxy formation is likely to persist in a more detailed model.

We show in Fig. 7 the comoving star formation rate we obtain for the three scenarios. We clearly see the inhibition of star formation as compared to the case with photoionization heating only. This also shows at large redshift [FORMULA], where new galaxies appear but the high entropy of the gas prevents the formation of the smallest galaxies which occur in the case with [FORMULA]. As explained in Sect. 2.5.3 this feedback effect is larger at low z where structure formation is further developped (hence the energy source is high) and the baryonic density is lower (hence cooling is less efficient). Moreover, this feedback effect onto star formation may partly explain the sharp drop of the star formation rate observed at low redshift [FORMULA]. Indeed, this decline is not very easy to obtain in usual models since at [FORMULA] the fraction of baryons which has been converted into stars is still very small [FORMULA]. As a consequence, it is difficult to get a sudden stop of the star formation process because there is plenty of gas available (although one can obtain in a natural fashion a decline as shown by the dotted line). On the other hand, the supernova or quasar heating of the IGM is able to suddenly change the conditions of star formation when the entropy of the gas becomes of the order of the entropy generated by gravitational collapse within the new non-linear halos.

[FIGURE] Fig. 7. The redshift evolution of the comoving star formation rate. The dotted line corresponds to photoionization heating only, the solid line to supernova heating (SN) and the dashed line to quasar heating (QSO). The data points are from Madau (1999), see references therein.

Although the shape of the redshift evolution of the comoving star formation rate we obtain for the quasar scenario (dashed line) agrees with observations, its normalization is somewhat too low. This might be the sign of a shortcoming of our description, in particular the assumption that reheating is uniform could lead to an overestimate of the entropy feedback onto star formation, since in a more realistic model with inhomogeneous reheating this effect may require more time to affect all the gas (especially for the small galaxies which form far away from clusters or groups of large galaxies and quasars). In fact, the star formation rate for the photoionization only case is already a bit two low at [FORMULA]. Thus our model of star formation is not perfect yet and we might underestimate the supernovae heating by a factor 2. Although this would translate into a supernovae efficiency factor smaller than unity, [FORMULA], this value remains quite large and it does not modify our conclusions (e.g. that the IGM is more likely to have been reheated by quasars). Thus, we think these results are already quite encouraging, in view of the simplicity of our model. On the other hand, note that we obtained a correct amplitude for the UV flux as shown in Fig. 4.

We show in Fig. 8 the fraction of matter enclosed within cooled objects (galaxies) and stars ([FORMULA]). We see that our results agree with the observed mass of stars (which may be more robust than the observed redshift evolution of the star formation rate itself). This could be expected since our model for galaxy formation (though in a more detailed version) was already checked in Valageas & Schaeffer (1999a) against observations. In particular, the cooled fraction [FORMULA] at [FORMULA] obtained with photoionization heating only (upper panel) is rather large. However, the mass of stars we obtain agrees with observations which shows that we do not encounter the overcooling problem. This is due to the small star formation efficiency implied by several processes: ejection of matter by supernovae and stellar winds, energy released by halos mergings and collapse.

[FIGURE] Fig. 8. The redshift evolution of the fraction of baryonic matter embedded within cooled objects [FORMULA] (solid line) and stars [FORMULA] (dashed line). The data point at [FORMULA] shows the observed mass within stars, from Fukugita et al.(1998).

Of course, we recover the behaviour seen in Fig. 7. In particular, for strong heating of the IGM we clearly see the saturation of the mass of cooled gas at lowzand the upper bound for the mass of stars it implies . Note that this feedback effect is quite general. Indeed, we require the reheating of the IGM to be large enough to affect cluster formation at [FORMULA] for halos with a virial temperature [FORMULA] keV. It is clear that this implies a significant effect onto galaxy formation at low redshift, since galaxies consist of shallower potential wells [FORMULA] keV. Since we must simultaneously describe galaxies, quasars and clusters, this leads to contradictory constraints. Indeed, the cluster [FORMULA] relation requires a high reheating temperature [FORMULA] K (see Sect. 4.3) while the observed star formation rate requires a small enough reheating, [FORMULA] K, so as not to inhibit too much galaxy formation. Thus, we see from Fig. 7 and Fig. 11 that these constraints imply a reheating temperature [FORMULA] K and [FORMULA]. This clearly shows the importance of using global models like ours (even though simplified) which allow one to evaluate the consequences of such processes on all objects (galaxies, clusters,...). Indeed, this provides strong constraints on such descriptions and it is the only way to test the global validity of these scenarios which should simultaneously account for all structure formation processes. We also note that a large fraction of baryonic matter at [FORMULA] is embedded within density fluctuations in the IGM and Lyman-[FORMULA] clouds with a rather large temperature [FORMULA] K. Part of this component may be difficult to observe.

4.2. Quasar luminosity function

We display in Fig. 9 the redshift evolution of the quasar luminosity function for all scenarios. We can see that at large redshifts [FORMULA] all curves nearly superpose since the additional energy source (from quasars or supernovae) plays no role at early times. On the other hand, as was the case for star formation, at small redshift [FORMULA] the entropy "floor" induced by quasar heating leads to a decrease of the quasar luminosity function which slightly improves the agreement with observations. Thus, although the decline at low z of the QSO multiplicity function is a natural outcome of our model even for the original scenario with photoionization heating only (dotted line), part of this decrease may also be due to the entropy production by quasars at earlier times. In view of the simplicity of our model for quasar formation, which is a natural outcome of our description of galaxies, we think our predictions agree reasonably well with observations. Of course, the physics of quasars may be more intricate than the description we use in this article but any meaningfull improvement would require a detailed model of the accretion processes leading to quasar formation, which is beyond the scope of this study.

[FIGURE] Fig. 9. The evolution with redshift of the B-band quasar luminosity function in comoving Mpc-3. The dotted lines correspond to photoionization heating only, the solid lines to supernova heating and the dashed lines to quasar heating. The data points are from Pei (1995).

4.3. Clusters

As we noticed in the introduction, the reheating of the IGM by supernovae or quasars at [FORMULA] can affect the formation of clusters at lower redshifts [FORMULA] since it leads to a mimimum entropy [FORMULA] of the gas which can break the expected scaling of the relation temperature - X-ray luminosity of clusters. In order to obtain an estimate of the characteristic virial temperature [FORMULA] where this transition should occur we define:


where we used (32). This is the temperature of the gas at the density contrast [FORMULA] (which defines clusters in our model) with the mean entropy of the IGM [FORMULA]. Thus, massive clusters with [FORMULA] are not affected by the entropy floor of the IGM because shock heating during the gravitational collapse of the halo generates a much larger entropy so that the usual scaling law is recovered. On the other hand, within smaller potential wells the gas has a larger entropy than the one produced by shock heating which leads to a smoother gas density profile and to a smaller density. This implies a smaller luminosity since [FORMULA]. In (40) we used the virial density rather than the core density, which is significantly larger, because most of the mass is characterized by densities of order [FORMULA] and we assume isothermal equilibrium. However, it is clear that (40) is only a rough estimate which could be uncertain by a factor 2.

We show in Fig. 10 the redshift evolution of [FORMULA]. Again, we can check that in the quasar scenario (QSO) the heating of the IGM occurs earlier than for the supernova case (SN). Moreover, since the quasar luminosity function drops at low z so that adiabatic cooling becomes the main process, as seen in Fig. 3, [FORMULA] decreases at low z. On the other hand, for the supernova scenario the fact that the redshift [FORMULA] is smaller leads to a smoother evolution at [FORMULA]. In any case, we see that we obtain at small redshift a characteristic temperature [FORMULA] keV. This is similar to the values which are used in studies of cluster formation (Cavaliere et al. 1997 use [FORMULA] keV while Valageas & Schaeffer 1999b use [FORMULA] keV). Indeed, this is how we chose the parameters [FORMULA]. Of course, smaller efficiency factors than those set in (37) lead to smaller IGM entropy and lower [FORMULA]. However, it is interesting to note that a larger [FORMULA] does not change much our results at [FORMULA] for the quasar scenario (QSO) because of the feedback effect we described above. On the other hand, for the supernova scenario (SN) the available range is already limited by the upper bound [FORMULA].

[FIGURE] Fig. 10. Redshift evolution of the characteristic temperature [FORMULA] which describes the influence of reheating on cluster formation. The dotted line corresponds to photoionization heating only, the solid line to supernova heating and the dashed line to quasar heating.

From the mean entropy of the IGM, or the characteristic temperature [FORMULA], we can obtain the cluster temperature - X-ray luminosity relation, as in Valageas & Schaeffer (1999b). The bolometric X-ray luminosity [FORMULA] of a cluster of volume V is:


where [FORMULA] is the electron number density and [FORMULA] is the bremsstrahlung emissivity function (in erg cm3 s-1) at temperature [FORMULA]. Thus, contrary to the Sunyaev-Zel'dovich effect, see (39), the X-ray luminosity strongly depends on the density profile of the hot gas within the cluster. During the gravitational collapse of the cluster shocks heat the gas up to the virial temperature T of the dark matter halo. However, the adiabatic compression of the gas from the IGM also heats the gas up to [FORMULA] which is a lower bound for the gas temperature [FORMULA]. In order to take into account both of these effects, we simply write for the final temperature of the gas:


Next, in order to obtain the density profile of the gas within the dark matter potential well we assume isothermal and hydrostatic equilibrium at the temperature [FORMULA] and we obtain (see also Cavaliere & Fusco-Femiano 1978):


where we used an isothermal profile [FORMULA] for the dark matter halo. Thus, for deep potential wells [FORMULA] we have [FORMULA] and the gas follows the dark matter density profile while for cool clusters [FORMULA] we get [FORMULA] and the gas density profile is smoother than the dark matter distribution. This change of the shape of the gas density profile leads to a break in the relation [FORMULA]. Moreover, in the inner parts of the cluster the density is large enough to lead to a small cooling time so that a cooling flow develops and some of the gas forms a cold component which does not emit in X-ray any longer. Thus, we define the cooling radius [FORMULA] as the point where the gas density reaches the threshold [FORMULA] such that:


where [FORMULA] is the cooling function (which is dominated by bremsstrahlung cooling for [FORMULA] keV) and [FORMULA] is the Hubble time. At large radii [FORMULA] the density is lower than [FORMULA] hence the local cooling time is larger than the Hubble time. Then, the gas distribution and the temperature had not had time to evolve much and the X-ray emissivity is proportional to [FORMULA], see (41). On the other hand, within the cooling radius [FORMULA] the gas had time to cool and form dense cold clouds. However, we consider that some of the gas is still hot and emits in X-ray as cooling does not proceed in a uniform fashion (Nulsen 1986; Teyssier et al. 1997). The density of this remaining gas has to be of order [FORMULA] and we write the X-ray luminosity of the cluster as:


where the factor 1 describes the contribution of the core, within [FORMULA], and the second term comes from the halo (note that the contribution within [FORMULA] is never much larger than the one from [FORMULA]). The factor [FORMULA] is a parameter of order unity which we use to normalize our relation to observations for massive clusters ([FORMULA] keV). We expect [FORMULA] which is indeed the case. It measures the density fluctuations of the gas distribution, since [FORMULA] and at any radius [FORMULA]. Note that our description is similar to the model developed by Cavaliere et al.(1997,1998) to describe the relation between the gas and the dark matter density profile, which shows in the quantity [FORMULA]. However, while they define a core radius from the density distribution itself (because they use a dark matter profile which grows more slowly than [FORMULA] at small r) the core radius we use describes the cooling of the gas, independently of the profile of the underlying dark matter halo. Using a shallower dark matter density profile would give similar results with a slightly larger [FORMULA] and [FORMULA] (see also Valageas & Schaeffer 1999b).

We show our results in Fig. 11 at [FORMULA] and [FORMULA], for both (SN) and (QSO) scenarios, using the redshift evolution we obtained in Fig. 10 for [FORMULA]. First, we can check that our results agree with observations for hot clusters [FORMULA] keV. Then, we see that the initial entropy of the IGM leads to a break of the relation [FORMULA] at low temperatures. However, for the supernova heating scenario we would require [FORMULA] to get a sufficiently large effect (though even for [FORMULA] we already see a knee in the relation [FORMULA]). On the other hand, for the quasar heating case we need [FORMULA]. Note that the strong redshift evolution of [FORMULA] in the quasar scenario, seen in Fig. 10, leads to a clear redshift dependence of the break of the relation [FORMULA]. On the contrary, the slow redshift evolution of [FORMULA] in the supernova scenario leads to a smoother redshift dependence of the relation [FORMULA]. This suggests that observations of the evolution of the temperature - luminosity relation at low [FORMULA] could provide some contraints on the (QSO) scenario. On the other hand, at large [FORMULA] there is almost no redshift evolution, which is consistent with observations (Mushotzky & Scharf 1997). Finally, we note that we may underestimate the effect of the heating from supernovae or quasars onto cluster formation in our model. Indeed, while we assumed this energy source to be homogeneous the gas which will later build a cluster is more likely to be reheated than an average calculation would show since clusters form at density peaks where the local density of galaxies and quasars is higher than average. This means that the break of the relation [FORMULA] could appear at a slightly larger temperature than shown in Fig. 10. This might "help" the supernova scenario as an efficiency factor [FORMULA] smaller than unity could be sufficient. However, it would probably remain close to [FORMULA]. On the other hand, we note that our constraints for the reheating of the IGM do not depend much on the details of the model of clusters since in any case in order to get a break of the [FORMULA] relation at [FORMULA] keV one necessarily needs to introduce a characteristic temperature [FORMULA] keV (Fig. 10) which sets the location of this bend. A more detailed model of clusters is presented in Valageas & Schaeffer (1999b) where a good match with observations is obtained with [FORMULA] keV.

[FIGURE] Fig. 11. The cluster temperature - X-ray luminosity relation at [FORMULA] (solid lines) and [FORMULA] (dashed lines). The upper panel shows the supernova scenario (SN) and the lower panel the quasar scenario (QSO). The data points are from Mushotzky & Scharf (1997) for clusters and from Ponman et al.(1996) for groups.

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Online publication: October 14, 1999