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

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3. Numerical results

We can now use the model we described in the previous sections to obtain the entropy history of the universe, as well as a consistent description of its reheating and reionization, together with the formation of quasars, galaxies and Lyman-[FORMULA] clouds. We shall 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/Mpc.

We shall consider three cases for the "entropy scenario". First, for reference we present some results we obtain for [FORMULA], as in VS. Then, we study both cases where only one of these two efficiency factors is non-zero. This allows us to see clearly the influence of each of these processes, as well as the magnitude of the relevant parameter [FORMULA] needed to get an appreciable effect. More precisely, in both cases we choose the value of [FORMULA] such that the mean IGM entropy [FORMULA] defined in (26) which we obtain at [FORMULA] satisfies [FORMULA]. Indeed, Ponman et al.(1998) find that the "entropy" [FORMULA] of small cool clusters seems to depart from the expected scaling law and to converge towards a floor value [FORMULA] keV cm2. This provides an upper bound for the IGM entropy [FORMULA] since these objects have just formed and we can expect the entropy of the hot gas to increase (e.g. through shocks) rather than decrease during gravitational collapse (before it cools). Moreover, if we assume that supernovae or quasars are indeed the source of this entropy floor (since gravitational collapse effects shoud not break the expected scaling law) this gives the value of the corresponding parameter [FORMULA]. In the actual universe it might happen that both sources of heating, supernovae and quasars, have the same magnitude. Then, the parameters [FORMULA] and [FORMULA] would be close to those we obtained for the individual cases. However, such a coincidence would be somewhat surprising.

3.1. Reheating of the universe

We show in Fig. 1 the reheating history we obtain for the three cases. The temperature [FORMULA] is a mass-averaged temperature which takes into account all the matter, from "voids" up to filaments, galaxies and clusters. The decrease with time of the IGM temperature at large redshift [FORMULA] ([FORMULA]) is due to the adiabatic expansion of the universe. Then, at [FORMULA] the IGM is slowly reheated by stars and quasars until it reaches at [FORMULA] a maximum temperature [FORMULA] K where collisional excitation cooling prevents any further increase. Eventually, at low z the temperature starts decreasing again because of the adiabatic expansion of the universe since the heating time becomes larger than the Hubble time (see upper panel in Fig. 3 below and Sect. 8.2 in VS). As described in Sect. 2.4 and Valageas et al.(1999a), at low z we set the characteristic temperature of Lyman-[FORMULA] clouds to [FORMULA] K when [FORMULA] becomes smaller than this value. Indeed, although the density contrast of small clouds can be small and even negative (their "overdensity" [FORMULA] can reach the minimum [FORMULA] shown in Fig. 5 below) they correspond to filaments or underdense objects, surrounded by regions of even lower density, which have decoupled from the expansion of the universe. Hence they do not cool because of adiabatic expansion (or at least this term is much smaller than for the IGM). In particular, we stress that on strongly non-linear scales even objects with a density contrast [FORMULA] equal to 0 are not described by the linear theory . Thus, the average density [FORMULA] loses the significance it has on large scales in the sense that it does not define any longer a density boundary between two physically different regimes. In fact, this role is now played by the density contrast [FORMULA], defined in (24), which describes most of the volume of the universe seen at small scales (see Valageas & Schaeffer 1997). In these regions, patches of matter with the density [FORMULA] appear as density peaks. This is clearly seen in Valageas et al.(1999b) where a comparison with numerical simulations shows that the same formulation (3), which is based on a stable-clustering approximation in a statistical sense, provides a reasonable description of objects defined by a density contrast which can vary from [FORMULA] downto [FORMULA].

[FIGURE] Fig. 1. The redshift evolution of the characteristic temperatures of the universe. We display the IGM temperature [FORMULA] (solid curve) and the virial temperature [FORMULA] (dashed line) of the smallest galaxies. The upper panel corresponds to [FORMULA] (only photoionization heating), the middle panel labelled "SN" to supernova heating ([FORMULA], [FORMULA]) and the lower panel labelled "QSO" to quasar heating ([FORMULA], [FORMULA]). The vertical line in both lower figures shows the redshift [FORMULA]. In the upper panel we also show the mass-averaged temperature [FORMULA] (dot-dashed line) and the mean temperature [FORMULA] (dotted line) of matter located outside galaxies and clusters.

As compared to the original model with only photoionization heating (upper figure) the main difference when we include a source term corresponding to supernovae (middle figure) or quasars (lower figure) is that the IGM temperature keeps increasing until [FORMULA] to reach a value [FORMULA] K (the decline of [FORMULA] at low z in the original case, and for the quasar scenario at [FORMULA], is due to adiabatic cooling, because of the expansion of the universe). In this case, as seen in (12) we take the Lyman-[FORMULA] clouds to be heated to the same temperature in order to have a self-consistent model. However, this high temperature could make it difficult to recover the observed properties of Lyman-[FORMULA] clouds at low z. Although this would deserve a detailed study we do not further investigate this point in this article where we mainly consider the energy requirements implied by efficient reheating. Moreover, the opacity due to Lyman-[FORMULA] clouds at low z does not influence much the reionization and reheating history of the universe (the medium is optically thin as shown by the Gunn-Peterson test). Besides, in order to get reliable estimates of the properties of Lyman-[FORMULA] clouds in the case of strong reheating by supernovae or QSOs one would certainly need to take into account the spatial inhomogeneities of this reheating process, which is beyond the scope of the present study.

In order to obtain the same entropy and temperature at [FORMULA] for the IGM, we see that the redshift [FORMULA] (shown by the dashed vertical line), where no just-virialized halo can cool, is higher for the quasar scenario (QSO) than for supernova heating (SN). Indeed, we obtain:


This is due to the peak of the quasar luminosity function at [FORMULA] and its sharp decline at lower redshift. Indeed, this implies that most of the heating process occurs at [FORMULA]. On the other hand, since the galaxy luminosity function evolves more slowly and does not drop at [FORMULA], the heating process due to stars keeps going on until [FORMULA] so that it appears delayed as compared to the quasar scenario, see discussion in Sect. 2.3. This also shows in the behaviour of [FORMULA] which keeps strongly increasing until [FORMULA] for supernova heating while it remains roughly constant (and even shows a slight decline) after [FORMULA] for quasar heating. The efficiency factors [FORMULA] and [FORMULA] we use are:


We can check that they are consistent with the estimates (18) and (19) and the requirement that [FORMULA] K at [FORMULA]. This latter value for [FORMULA] is due to the constraint [FORMULA] which we impose in order to get the break at [FORMULA] keV of the relation [FORMULA] for clusters, as explained above and described in details in Sect. 4.3. In fact, for supernova heating the efficiency factor [FORMULA] is not sufficient (though not by far since [FORMULA] is enough) to explain the break of the cluster [FORMULA] relation, as discussed in Sect. 4.3. However, since we must have [FORMULA] we keep this value for the supernova efficiency factor. On the other hand, we note that a very small value for the quasar efficiency factor [FORMULA] is sufficient to raise the entropy of the IGM to a level high enough to explain the cluster observations. Note also that [FORMULA], where [FORMULA] defined in Sect. 2.3 measures the quasar radiative efficiency. Thus, the quasar scenario (QSO) appears to be quite reasonable while the supernova hypothesis (SN) seems less likely. However, further work is needed in order to assess with a sufficiently good accuracy the efficiency of these two processes of energy transfer before one can draw definite conclusions.

3.2. Entropy production

We display in Fig. 2 the redshift evolution of the entropy of the gas. It increases with time as structure formation develops and heating processes grow. The specific entropy [FORMULA] of underdense regions is larger than the mean IGM entropy [FORMULA], and increasingly so at smaller redshift, because of their low density [FORMULA]. As was the case for the temperature evolution, we note that the entropy shows a faster rise at low z for the supernova scenario (SN) than for the quasar heating (QSO). Again this is due to the fact that the quasar energy output (proportional to the luminosity function) peaks at [FORMULA] contrary to the galaxy contribution which keeps increasing until [FORMULA]. The entropy [FORMULA] is defined from the temperature [FORMULA] and the density contrast [FORMULA]. The fact that it is lower than [FORMULA], especially at low z, shows that the smallest galaxies are influenced by the entropy floor set by the IGM. However, we recall that [FORMULA] is not the entropy of the gas in such a galaxy, since the gas compression stops when it reaches the density contrast [FORMULA] and it subsequently cools and falls into the dark matter potential well, which diminishes its entropy (transfered into the radiation).

[FIGURE] Fig. 2. The redshift evolution of the characteristic entropies of the universe. We display the mean IGM entropy [FORMULA] (dot-dashed curve), the entropy [FORMULA] of underdense regions (solid line) and the entropy which would correspond to the smallest galaxy [FORMULA] (low dashed line). From top downto bottom, the various panels correspond to photoionization heating only, supernova heating and quasar heating, as in Fig. 1.

We note that if we could measure the temperature and the entropy of the gas within Lyman-[FORMULA] clouds or small groups at high redshift [FORMULA] we might be able to see which scenario, (SN) or (QSO), is best favoured, using the fact that the redshift evolution is slower for the quasar heating process. However, it is clear that this would not give a definite answer because of the uncertainty associated to these poorly known processes.

3.3. Time-scales

We display in Fig. 3 the redshift evolution of the various characteristic time-scales. For the original model (upper panel), at large redshift [FORMULA] the smallest time-scale is the photoionization heating time, defined in (14), which means that the IGM temperature increases in this redshift range (see Fig. 1). At lower and larger redshift, the smallest time is the Hubble time [FORMULA] (corresponding to the ordinate 0 in the figure) which implies that the IGM cools due to the adiabatic expansion. The dashed curve which shows a peak at [FORMULA] corresponds to atomic processes within the IGM (collisional excitation, collisional ionization, bremsstrahlung, Compton cooling or heating,...). Most of the time it is dominated by Compton cooling or heating which explains the peak at [FORMULA] when [FORMULA] ([FORMULA]). At higher redshift the IGM gas is heated by CMB photons while at lower z the IGM is cooled through the interaction with the CMB. Around reionization at [FORMULA] the main process is collisional excitation (see VS for details). In both lower panels we also display the heating time-scale [FORMULA] or [FORMULA] associated to the additional energy source. We can see that it becomes the dominant process somewhat after reionization at [FORMULA]. Then, since [FORMULA] becomes the smallest time-scale the corresponding energy source heats the IGM up to [FORMULA] K. We can note again that the supernova heating is somewhat delayed as compared to the quasar scenario. The rise at low z of [FORMULA] as compared to the upper panel is due to the growth of the IGM temperature, see (14).

[FIGURE] Fig. 3. The redshift evolution of the characteristic heating and cooling time-scales. All times are shown in units of the Hubble time [FORMULA]. The dashed curve shows the cooling or heating time associated to atomic processes (Compton cooling, collisional excitation,...). The solid line labelled [FORMULA] shows the photoionization heating time. The dot-dashed (resp. dotted) line labelled [FORMULA] (resp. [FORMULA]) shows the heating time-scale associated to supernovae (resp. quasars). From top downto bottom, the various panels correspond to photoionization heating only, supernova heating and quasar heating, as in Fig. 1.

Note that in all three cases the IGM is reheated and reionized by the energy output of galaxies and QSOs (whether it is radiation or kinetic energy). Thus in all scenarios we expect a proximity effect (along a line of sight to a distant QSO the IGM is more ionized close to the quasar) since these processes are not exactly homogeneous (they are more efficient close to the sources). This also holds for the (SN) scenario because QSOs are associated with galactic cores (where a black hole is surrounded by an accretion disk) which means that they are embedded within regions of star formation (which occurs in the host galaxy). In fact, this proximity effect provides a measure of the inhomogeneity of the reheating and reionization process. Note that Davidsen et al.(1996) and Reimers et al.(1997) do not observe any proximity effect which suggests that our homogeneous approximation is reasonable.

3.4. Reionization history

We show in Fig. 4 the redshift evolution of the background UV flux [FORMULA] defined by:


[FIGURE] Fig. 4. Redshift evolution of the background UV flux [FORMULA]. The dotted line corresponds to photoionization heating only, the solid line to supernova heating and the dashed line to quasar heating. 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 very sharp rise at [FORMULA] corresponds to the reionization redshift when the universe suddenly becomes optically thin. We can see that all scenarios give nearly the same results since at large redshift the entropy of the IGM is not sufficiently large to significantly affect galaxy and quasar formation. This was also apparent in Fig. 1 and Fig. 2. In particular, at high z the gas densities are large so that cooling is quite easy. As a consequence, the entropy of the IGM cannot prevent the gas from cooling and falling into galactic halos to form stars or quasars so that the reionization redshift does not depend on the efficiency factors [FORMULA] . This is in fact reassuring, since it shows that most of the results we obtained in VS (e.g. ionization state of hydrogen and helium,...) are still valid and do not depend on the injection of energy into the IGM by stars or quasars (the reionization redshift we obtain here is larger than in VS because we use [FORMULA] K instead of [FORMULA] K). The UV flux in the quasar scenario (QSO) is somewhat larger than for both other cases at [FORMULA] ([FORMULA]) although the star formation rate is a bit smaller (see Fig. 7 below) because the opacity of the universe is lower. Indeed, this implies a smaller absorption of the radiation emitted by stars and quasars. This is due to the larger temperature of the IGM (see Fig. 1) which leads to a lower opacity from Lyman-[FORMULA] clouds. However, this is only relevant for [FORMULA].

3.5. Characteristic density contrasts

We present in Fig. 5 the redshift evolution of the characteristic density contrasts within the universe. The lower solid line shows the "overdensity" [FORMULA] of the underdense regions which cover most of the volume, see (24). At large z we have [FORMULA] since very few baryonic structures have formed and the universe appears as a nearly uniform medium. At low z it declines and can reach very low values [FORMULA] as "voids" appear amid filaments and halos. It is larger for both lower panels because the IGM temperature is higher, which means that the scale [FORMULA] is larger, and dark-matter density fluctuations are smaller on larger scales (following the behaviour of the two-point correlation function). On the other hand, the mean density contrast [FORMULA], which takes into account all the matter which is not enclosed within galactic halos where the gas was able to cool, from "voids" up to filaments and forest Lyman-[FORMULA] clouds, remains close to unity. Indeed, the fraction of matter [FORMULA] which is not embedded within galactic halos remains large until [FORMULA] since we have [FORMULA] at [FORMULA], see Fig. 8. The mean [FORMULA] is larger than [FORMULA] since it is weighted by mass which gives more weight to dense regions (filaments, clouds) as compared to low-density areas. Finally, the density contrast [FORMULA] corresponds to the small galactic halos (which are not influenced by the cooling condition [FORMULA]). Thus, at all times in the upper panel and at [FORMULA] in the lower panels it is equal to the "virialization" density contrast [FORMULA] obtained from the usual spherical model. At low redshift in both lower panels it is larger than [FORMULA] and equal to the expression defined in (34) because of entropy considerations. Again, we note that for the quasar scenario (QSO) this effect appears somewhat earlier.

[FIGURE] Fig. 5. The redshift evolution of the characteristic density contrasts. We display the density contrast [FORMULA] of large underdense regions in the IGM (lower solid line), [FORMULA] for the mean IGM density (lower dashed line), [FORMULA] for the mass-averaged density within the IGM (upper dot-dashed line) and [FORMULA] for galactic halos (upper solid line).

3.6. Compton y parameter

The hot gas within the IGM or virialized halos (e.g. clusters) scatters some photons of the CMB from the low energy Rayleigh-Jeans part of the spectrum up to the high energy Wien tail. The magnitude of this perturbation is conveniently described by the Compton parameter y:


We consider three components to the global Sunyaev-Zeldovich effect. Thus we define [FORMULA] (noted [FORMULA] in VS) describing the effect of low-density regions, [FORMULA] which takes into account all the matter within the IGM (from voids up to filaments and Lyman-[FORMULA] forest clouds) and [FORMULA] which describes the effect from virialized halos above [FORMULA] (i.e. galaxies and clusters). Both [FORMULA] and [FORMULA] reach a plateau at [FORMULA] since at earlier times the universe was almost exactly neutral. On the other hand [FORMULA] saturates earlier because the fraction of matter embedded within massive virialized halos declines at large z. The contribution [FORMULA] of low-density regions is always small since it only contains a small fraction of matter at low z with a small temperature. Similarly, [FORMULA] from galaxies and clusters is much larger than [FORMULA] in the upper panel because although both components contain similar fractions of baryonic matter the temperature of these virialized halos is much larger than the IGM temperature (see Fig. 1). However, when there is an additional source of energy from supernovae or quasars [FORMULA] becomes non-negligible, especially for the quasar scenario (QSO) where [FORMULA] rises earlier. Note that on general grounds we can expect [FORMULA] to be at most of the same order as [FORMULA] since a large fraction of matter is embedded within virialized halos and the temperature of the IGM should not be larger than [FORMULA] K while the temperature of these collapsed objects can be much larger (e.g. clusters have [FORMULA] K). A more precise study of the Compton parameter induced by clusters is presented in Valageas & Schaeffer (1999b). In any case, we find that the Compton parameter from any contribution is much lower than the COBE/FIRAS upper limit [FORMULA] provided by observations (Fixsen et al. 1996). This means that we cannot constrain the entropy scenario from its effect on the Compton parameter using current observations. On the other hand, it shows that we do not contradict the data, hence these heating processes remain plausible explanations for the cluster observations.

We can also compute the X-ray background provided by the IGM, from underdense regions up to Lyman-[FORMULA] clouds in filaments. However, we find that it is negligible since we get at most (for the QSO scenario) a flux in the 0.5-2 keV band of [FORMULA] erg s-1 cm-2 deg-2 from the IGM, and [FORMULA] erg s-1 cm- 2 deg-2 from virialized halos which have not cooled, as compared to the observed extragalactic intensity [FORMULA] erg s-1 cm- 2 deg-2 (Miyaji et al. 1998). On the other hand, the contribution from galaxies and clusters is of the order of [FORMULA] erg s-1 cm- 2 deg-2. Indeed, the X-ray flux is very sensitive to the density (it varies as [FORMULA]) and to the temperature. Thus, most of the contribution comes from high temperature clusters [FORMULA] keV while the low temperature of the IGM [FORMULA] K [FORMULA] keV implies a very small contribution since [FORMULA].

[FIGURE] Fig. 6. The Compton parameter y up to redshift z describing the Sunyaev-Zeldovich effect from the "voids": [FORMULA] (lower dashed line), the IGM as a whole (voids, filaments and clouds): [FORMULA] (solid line) and virialized halos: [FORMULA] (upper dashed line).

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© European Southern Observatory (ESO) 1999

Online publication: October 14, 1999