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Astron. Astrophys. 345, 329-362 (1999)

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5. Time-evolution

The method we presented in the previous sections can obviously be used to derive the galaxy properties at any time, which allows us to get the evolution with redshift of the physical characteristics of galaxies and of their mass function or luminosity function.

5.1. [FORMULA]

5.1.1. Dark matter properties

The relation temperature-density which corresponds to Fig. 2 is shown in Fig. 12 at the redshifts [FORMULA] and [FORMULA]. Its behaviour is similar to the one we got at [FORMULA]. As we noticed in Sect. 3.2 the curves [FORMULA], hence [FORMULA], depend on z. This changes the shape of [FORMULA], which will modify the shape of the luminosity function [FORMULA] as we shall see.

[FIGURE] Fig. 12. Temperature-density diagram in the case [FORMULA] at the redshifts [FORMULA] from top down to bottom. The critical density at [FORMULA] is noted [FORMULA]. As in Fig. 3, the short-dashed lines correspond to halos of fixed mass, on the left, and halos of fixed radius, on the right. Their position does not vary with the redshift.

We can notice in Fig. 12 that as the redshift increases halos get smaller and less massive. However, since [FORMULA] does not change with z some halos have fixed physical characteristics below a certain redshift, as long as they stay on [FORMULA] (in the approximation of stable clustering, which holds for [FORMULA]). We may consider the evolution of halos in the extreme regimes 1) and 3):

1) Faint galaxies located on [FORMULA] constant.

At fixed temperature T we get [FORMULA] and [FORMULA]. These halos become smaller and less massive, and we have:

[EQUATION]

where we assumed stable clustering. Thus, as long as the exponential cutoff plays no role ([FORMULA] or [FORMULA]), the temperature function given by the PS approach verifies [FORMULA], that is [FORMULA] if [FORMULA], while the non-linear scaling approach gives [FORMULA] if [FORMULA] and [FORMULA]. Hence the comoving number density of halos at these small temperatures increases with z in both prescriptions, but somewhat more strongly within the framework of the PS approach. However, since [FORMULA] and x increase the exponential cutoff will eventually lead to a decrease of the comoving number density.

3) Bright galaxies, located on [FORMULA] constant

The physical properties of these halos remain constant with the redshift while their density contrast decrease as [FORMULA]. This holds as long as [FORMULA]. Their parameters [FORMULA] and x (if the clustering is stable) remain constant too so the temperature function does not evolve. This implies that the changes of the comoving luminosity function in the range corresponding to these halos will only be due to the variation of their luminosities (pure luminosity evolution).

As the redshift increases, the regime of galaxies corresponding to a fixed temperature changes: [FORMULA]. Finally, for [FORMULA] all halos belong to the regime 1).

5.1.2. Star formation history

Fig. 13 shows the star formation rate [FORMULA] for various redshifts as a function of [FORMULA]. Small galaxies, with a low circular velocity, have a small star formation rate because their star formation time-scale [FORMULA] is very long ([FORMULA]) since their density and temperature are small. On the other hand, very large and luminous galaxies also have a small star formation rate in the present universe because their star formation time-scale is short as compared to their age ([FORMULA]), so they have already consumed most of their gas. In the past they had a higher star formation rate since their gas content was larger.

[FIGURE] Fig. 13. Star formation rate in [FORMULA]year as a function of the circular velocity [FORMULA] in km/s, for the redshifts [FORMULA] from top down to bottom.

More precisely, we get for the regimes 1) and 3):

1) Faint galaxies located on [FORMULA]:

We have [FORMULA] and [FORMULA]. Since [FORMULA] we obtain [FORMULA].

3) Bright galaxies, located on [FORMULA]:

For galaxies which have already consumed most of their gas content we have [FORMULA] and [FORMULA]. Hence [FORMULA].

Fig. 14 shows the time-evolution of the star formation rate [FORMULA] (upper panel) and of the metallicities [FORMULA] and [FORMULA] (lower panel) of a galaxy similar to the Milky Way. The slight decrease with time of the star formation rate after [FORMULA] years is due to the smaller gas content of the Galaxy, as its gas is gradually converted into stars. This variation by about a factor 2 is consistent with observations, which imply in fact that the star formation rate of the Milky Way did not vary by much more than this amount. Thus, another prescription with a stronger evolution would increase the global comoving star formation rates or the luminosity functions we shall obtain at [FORMULA], which would improve the agreement with observations, but it would not satisfy the mild evolution observed for the Milky Way. One could build for instance such a scenario by using for the star formation time-scale [FORMULA] a different prescription: [FORMULA] (where [FORMULA] is the gas density in the core) instead of [FORMULA]. Such a parameterization, quite plausible, leads naturally to a stronger dependence on time, through the density, and to the effects we described above, as we checked numerically. To keep things simple we shall keep our original prescription, which satisfies the constraints given by the Milky Way. The slight increase of the star formation rate shortly after the time [FORMULA] is due to the fact that some gas is still falling onto the galaxy inner parts from the halo.

[FIGURE] Fig. 14. Upper figure: evolution with time of the star formation rate [FORMULA] of the Milky Way. Lower figure: evolution of the metallicities [FORMULA] and [FORMULA] of the Milky Way. The vertical dashed-line represents the time [FORMULA]. The marks on the upper left side of both figures show the redshift.

The temporary increase with redshift of the star formation rate at small times [FORMULA] (on the left of the vertical dashed-line) is due to the decrease of the relevant time-scales [FORMULA] and [FORMULA] at high redshift: [FORMULA]. Note however that during this epoch the mass which will later form the Milky Way is distributed over several smaller halos. The metallicities (meant as [O/H] rather than [Fe/H] since we do not include SNIs) increase steadily with time, in all three components ([FORMULA] and [FORMULA]). At the time [FORMULA], the metallicities [FORMULA] and [FORMULA] were smaller than their present values by a factor [FORMULA], which agrees with the difference in metallicity between the oldest and youngest stars in the disk. Stars created before this date with a high metallicity [FORMULA] are located within the bulge, while even older and less metallic stars would be located in the halo. The very steep increase in metallicity at the begining implies that there are very few stars of very low metallicity [Z][FORMULA].

Fig. 15 shows the cumulative stellar metallicity distribution [FORMULA] for the Milky Way:

[EQUATION]

Thus, at time [FORMULA], which corresponds to the formation of the disk, the metallicity of the gas, and of the stars being formed, was a factor 0.3 smaller than the metallicity of stars created today. Moreover, [FORMULA] of present long-lived stars were formed by this date, which is roughly the stellar mass of the bulge. Thus, in our model bulges of disk galaxies contain an important proportion of old stars (with [FORMULA] for the Milky Way) which formed before the disk along processes similar to those of elliptical galaxies. This agrees with the interpretations of Ortolani et al. (1995) (who found bulge globular clusters as old as halo globular clusters) and of Jablonka et al. (1996). Note however that some amount of star formation may have kept going on until today. The slow decrease of the metallicity in Fig. 15 with the fraction of the mass of stars also shows that we have no G-dwarf problem.

[FIGURE] Fig. 15. The cumulative stellar metallicity distribution for the Milky Way. The vertical dashed-line represents the metallicity of the newly formed stars at the time [FORMULA], when the object switches from the merging regime to the isolated galaxy regime.

Fig. 16 shows the distribution function of the star formation rates at different redshifts (i.e. the number of galaxies with a given star formation rate per comoving Mpc3). It extended to much higher star formation rates in the past, at [FORMULA], than it does in the present universe, because massive galaxies formed most of their stars at these early epochs (since they have a small star formation time-scale), when they experienced a very active phase, and their star formation rate has steadily declined ever since as their gas content became smaller. This corresponds to the history we developed previously in detail for a galaxy similar to the Milky Way. Moreover, the luminosity (or the galactic mass) of the galaxies characterized by the highest star formation rate was larger in the past (at [FORMULA]) than it is now. These results agree with the redshift evolution observed by Cowie et al. (1996, 1997).

[FIGURE] Fig. 16. The distribution function of the star formation rates at different redshifts: [FORMULA] (solid line), [FORMULA] (dashed line), [FORMULA] (dot-dashed line) and [FORMULA] (dotted line).

5.1.3. Luminosity evolution

We can see in Fig. 17 that when the redshift increases short-lived stars become more important because these halos are younger, have more gas and a higher star formation rate. Hence the global mass-luminosity ratio gets smaller and, at fixed [FORMULA], massive halos which are on the cooling curve [FORMULA] have a larger luminosity since their mass remains constant (as long as they remain on [FORMULA]). Thus, the slope of the temperature-luminosity relation at high [FORMULA] gets stronger. As a consequence, the knee of the luminosity function should move toward larger luminosities in the past, and fainter luminosities in the future, since we noticed above that the temperature function in this region does not evolve with z. On the other hand, at small [FORMULA] the temperature-luminosity relation keeps the same slope since the analysis developed for [FORMULA] is still valid.

[FIGURE] Fig. 17. Upper figure: ratio of the total mass of stars [FORMULA] over the galaxy luminosity L, as a function of the circular velocity [FORMULA], for the redshifts [FORMULA] and [FORMULA]. Lower figure: luminosity L of the galaxy as a function of [FORMULA]. The dot-dashed curves show the luminosity due to massive stars ([FORMULA]). At large [FORMULA] a higher z corresponds to larger L and larger [FORMULA].

However, the normalization decreases at higher redshifts, although the mass/luminosity ratio is smaller, because the mass of these halos decreases. Indeed, at higher redshifts the time [FORMULA], which is the life-time of stars at the boundary between our two classes of stars (short-lived which are recycled and long-lived ones), scales as [FORMULA], see Appendix D. The mass/luminosity ratio of the global stellar population varies more slowly than [FORMULA] because the IMF contains fewer massive stars: [FORMULA] where x and [FORMULA] are stellar parameters (IMF, mass-luminosity relation, see Appendix D). Since for faint galaxies, located on [FORMULA], we have for a fixed temperature [FORMULA], because the global star formation time-scale [FORMULA] follows the decrease of the galactic age, the luminosity decreases slowly at high redshifts, as we can check in Fig. 17. As galaxies leave the regime 3) to enter the regimes 2) (even when they remain on [FORMULA], the change is that now their stellar content is small: [FORMULA] and [FORMULA]) and finally 1), they satisfy the scaling [FORMULA] described in Sect. 4.1.3 and Appendix G for the regime 1) at [FORMULA]. This explains why the slope of the Tully-Fisher relation we get remains constant with z, and extends up to the bright galaxies at [FORMULA] (the high luminosity bend we have for [FORMULA] disappears). The regime 1), which only corresponded to the smallest galaxies at [FORMULA] is now valid for nearly all galaxies at [FORMULA].

5.1.4. Luminosity function

Fig. 18 shows the evolution with redshift of the comoving galaxy luminosity function in the case of a CDM power-spectrum. We can see that the knee of the luminosity function moves toward larger luminosities in the past, until [FORMULA], and fainter luminosities in the future, while the comoving number density for faint galaxies increases until [FORMULA]. This is consistent with what we expected from the above analysis. Indeed, for faint galaxies we saw previously that [FORMULA] for the PS approach and [FORMULA] for the non-linear scaling prescription. Since the Tully-Fisher relation does not change very much in this range, the luminosity function [FORMULA] follows the same behaviour.

[FIGURE] Fig. 18. The comoving galaxy luminosity function [FORMULA] at the redshifts 5, 2, 1, 0 and -0.5, for the non-linear scaling approach (solid line) and the PS prescription (dashed line). The data points are observational results at [FORMULA] as in Fig. 9.

On the contrary, for bright galaxies we explained above that while the temperature function [FORMULA] does not evolve, the slope of the Tully-Fisher relation gets larger, so that a given temperature corresponds to a larger luminosity in the past, hence the luminosity function cutoff moves toward larger luminosities. We can also notice that the faint-end slope of the luminosity function gets higher in the past. As we said previously, this is related to the change of the curve [FORMULA] with z. Indeed, these faint galaxies leave the regime 2) to enter 1), hence the slope of the luminosity function increases from [FORMULA] to [FORMULA] for the PS approach, and from [FORMULA] to [FORMULA] for the non-linear scaling prescription (note that although this increase is qualitatively correct, as we can see in Fig. 18, the slopes of the luminosity function we obtained in this way are not very accurate because we did not consider the variation of the ratio [FORMULA]). This effect could explain the steepening of the faint-end slope of the luminosity function which is observed in the past at [FORMULA] or [FORMULA]. Moreover, the slope of the luminosity function after the cutoff gets smaller, especially for [FORMULA]. Indeed, we still have (with [FORMULA]):

[EQUATION]

but as galaxies leave the regime 3), where [FORMULA], to enter the regime 2), where [FORMULA], as their gas content increases, the mass-luminosity relation becomes [FORMULA] instead of [FORMULA]. Hence we get:

[EQUATION]

[EQUATION]

With [FORMULA] and [FORMULA] we obtain:

[EQUATION]

Thus, in both cases the exponential cutoff becomes less sharp in the past (note that the cutoff characteristics [FORMULA] or [FORMULA] are not the same for the PS and non-linear scaling approaches, as we noticed earlier, in Sect. 4.1.1. for instance).

We can see that the galaxy comoving number density evolves much faster for the non-linear scaling prescription than for the PS approach. Indeed, it leads to a luminosity function which is much higher than the PS one at the bright end for [FORMULA] and it suddenly decreases at [FORMULA] to superpose onto the PS prediction at [FORMULA] and then gives even fewer galaxies than this latter prescription. The slow evolution for the PS case is due to the fact that [FORMULA], hence [FORMULA], is constant in the regime 3), and only increases as [FORMULA] once the halo enters the domain 1). Moreover, when [FORMULA] is not too large, in this regime 1) the prefactor grows as [FORMULA] which balances the decrease of the exponential term. On the contrary, in the case of the non-linear scaling prescription x increases slowly until [FORMULA], since the clustering is not exactly stable. Then, when [FORMULA] the correlation function [FORMULA] decreases suddenly very strongly, and x rises sharply. This produces a sharp decrease of the comoving number density, as the exponential cutoff becomes very important. We must note that for [FORMULA] the function [FORMULA] should change so that for [FORMULA] it becomes equal to the result obtained in the quasi-gaussian regime (see Colombi et al. 1997). However, the exponential cutoff of the the quasi-gaussian function is stronger than for the non-linear case (Colombi et al. 1997), so the predicted decrease of the luminosity function would be somewhat stronger. Thus, it is important to note that up to [FORMULA], which is already a rather large redshift, the non-linear scaling approach leads to a much higher luminosity function than the PS prescription at the bright end. In this regime, [FORMULA] is still of the order of [FORMULA] at the onset of the exponential fall-of. At such values of [FORMULA] it has been seen (Bouchet et al. 1991) that [FORMULA] is still given to a good approximation (10%) by its non-linear form.

5.1.5. Average comoving stellar properties

The evolution with redshift of the stellar density parameter [FORMULA], see (C11), is displayed in Fig. 19 (upper panel). Its decrease in the past is due to two effects. First, we only consider halos with a temperature [FORMULA] K (cutoff due to inefficient cooling), which leads to higher parameters [FORMULA] and x at high redshifts, see (36), hence to a smaller mass fraction - there is less mass contained in deep potential wells in the past. Second, at larger redshifts galaxies have a higher gas/star mass ratio (smaller [FORMULA]), hence most of the mass is in the form of gas. Note that in the present universe, since only galaxies more luminous than the Milky Way have an appreciable stellar content (our Galaxy is just between both regimes as [FORMULA]) and they form a small part of the total mass (they already are in the exponential cutoff of the luminosity function) the ratio [FORMULA] is still small.

[FIGURE] Fig. 19. Upper figure: evolution with redshift of the stellar density parameter [FORMULA]. Lower figure: comoving star formation rate [FORMULA] for [FORMULA]. The solid lines correspond to the case [FORMULA] and the dashed lines to [FORMULA]. In both cases we only display the non-linear scaling prescription. The data points are taken from Madau et al. (1996) (squares), Lilly et al. (1996) (disks), Gallego et al. (1995) (triangle) and Cowie et al. (1995) (crosses). Note that the points at high redshift ([FORMULA]) are only lower limits.

We can see in the figure that two thirds of the present mass in stars formed recently at [FORMULA]. It is also clear that we do not encounter the usual overcooling problem - all the gas cools and is converted into stars within small objects at high redshifts because cooling is very efficient (high densities) - within this framework. As we explain in C.2, this is due to the redshift dependence of our star formation time-scale [FORMULA], which ensures that although the gas may cool at high redshift it cannot be immediately converted into stars ([FORMULA] does not decrease faster than the age of the universe with redshift). Hence there is still plenty of gas available in the present universe ([FORMULA]) which is not necessary cold as it has been reheated by supernovae, stellar winds and by the energy released by halo mergings and collapse.

The comoving star formation rate, see (C13), is shown in Fig. 19 (lower figure). It first increases with redshift until [FORMULA] because the star formation time-scale [FORMULA] decreases as [FORMULA], as long as [FORMULA]. However, at high redshifts [FORMULA] the comoving star formation rate gets smaller because the mass contained in deep potential wells starts to decrease and [FORMULA] gets larger because of the factor [FORMULA]. Nevertheless, its variation over the whole range [FORMULA] is rather small.

The evolution with redshift of the B-band comoving luminosity density is shown in Fig. 20. The luminosity density decreases in the past because the mass fraction within galaxies diminishes and their stellar content is lower. However, the luminosity density remains high until [FORMULA], when it is still equal to one half of its present value.

[FIGURE] Fig. 20. The B-band comoving luminosity density. The solid line corresponds to the case [FORMULA] and the dashed line to [FORMULA]. In both cases we only display the non-linear scaling prescription.

On all these figures, we can see that the evolution is slower for a low-density universe, as is well-known.

5.1.6. Galaxy counts

Finally, the evolution of the luminosity function allows us to calculate the galaxy number counts as a function of the apparent magnitude [FORMULA] and the redshift z. The absolute B band magnitude [FORMULA] is related to [FORMULA] by:

[EQUATION]

where [FORMULA] is the luminosity distance to redshift z, [FORMULA] is the usual K-correction and [FORMULA] is the evolution correction. These last two terms vary with the stellar and morphological properties of galaxies, and should be evaluated from our galaxy evolution model, to get a self-consistent result. However, since this would require a detailed description of the stellar properties of galaxies, including their spectra and colors, which we plan to tackle in a future paper, we shall simply take in this article [FORMULA]. Indeed, as can be seen in King & Ellis (1985) for instance, while [FORMULA] is positive because of the shape of the spectrum, [FORMULA] is negative because galaxies where bluer in the past, and both terms cancel roughly. Of course, the net result varies with the galaxy type, and the evolution model used to get [FORMULA], and for faint magnitudes where the contribution to galaxy counts extends to high redshifts ([FORMULA]) we may have an error of one magnitude.

However, this approximation should give a fair idea of the implications of our model on the galaxy number counts, and it does not influence the comparison between the PS and non-linear scaling approaches. We note [FORMULA] the number of galaxies per square degree with apparent magnitude [FORMULA] to [FORMULA] and redshift z to [FORMULA]:

[EQUATION]

where [FORMULA] is the comoving luminosity function at the redshift z. Finally, the number of galaxies [FORMULA] per square degree is the integral over z of [FORMULA]:

[EQUATION]

Thus, the PS approach leads to:

[EQUATION]

while the non-linear scaling prescription gives:

[EQUATION]

Fig. 21 shows the B counts for the PS and non-linear scaling prescriptions, as well as for a no-evolution model such that the comoving luminosity function does not change with time and is equal to the one given by the non-linear scaling prescription at [FORMULA], which allows to distinguish the effects of evolution from those due to geometry (comoving volume element, luminosity distance). Our results agree reasonably well with observations although the number counts given by the non-linear scaling model are somewhat too small at very faint magnitudes [FORMULA]. The counts given by the non-linear scaling approach are lower than those obtained by the non-evolving model because the number of [FORMULA] galaxies we get with our model decreases when we look in the past (see Fig. 18) and for a fixed apparent magnitude the counts are mainly sensitive to the evolution of the number density of bright galaxies, since we do not see any longer the faintest galaxies. This discrepancy with the observational data for [FORMULA] is a well-known problem for non-evolving models. In fact, Cole et al. (1992) showed that natural models where the comoving luminosity function evolves in a homogeneous way (evolution of the normalization or of the cutoff) so as to match the B counts will contradict the data in the K band and the observed redshift distribution of galaxies. Hence they concluded that a new population of rapidly evolving blue galaxies fainter than [FORMULA] is necessary to fit all the data. Such an effect, even if real, cannot be given by our simple parameterization of star formation. This point deserves a detailed study that will be done elsewhere.

[FIGURE] Fig. 21. The B band differential number counts. The graph shows [FORMULA] (number of galaxies per square degree per apparent magnitude) for the non-linear scaling approach (solid line), the PS prescription (small dashed line), and a no-evolution model (dot-dashed line) such that the comoving luminosity function does not change with z. The data points are taken from Lilly et al. (1991) (squares) and Metcalfe et al. (1991) (triangles).

Fig. 22 shows the redshift distribution [FORMULA] of galaxies selected at an apparent magnitude [FORMULA], for the PS and non-linear scaling prescriptions, and for the no-evolution model. Of course we recover the same features as for the integrated number counts: the PS prescription (because of its high normalisation) and the non-evolving model give more galaxies than the non-linear scaling approach. For these last two cases we find a peak at [FORMULA] which is consistent with observations (Cowie et al. 1996; Colless et al. 1993; Colless et al. 1990; Broadhurst et al. 1988) for this apparent magnitude. Note also the fast evolution of the PS result.

[FIGURE] Fig. 22. Redshift distribution of galaxies with apparent B band magnitude [FORMULA]. The graph shows [FORMULA] for the non-linear scaling approach (solid line), the PS prescription (small dashed line), and the no-evolution model (dot-dashed line). The data points (histogram) are from Cowie et al. (1996) for galaxies in the range [FORMULA].

5.2. [FORMULA]

In the case [FORMULA] and [FORMULA] the analysis developed for a critical universe still holds. Fig. 23 shows the evolution of the comoving galaxy luminosity function with the redshift in this case for a CDM power-spectrum. As we can see, the evolution is qualitatively similar to what we could see in Fig. 18 but much slower. This is a well-known property of low-density universes: the evolution of gravitational clustering is slower than for a critical universe.

[FIGURE] Fig. 23. The comoving galaxy luminosity function [FORMULA] at the redshifts 5, 2, 1, 0 and -0.5, for the non-linear scaling approach (solid line) and the PS prescription (dashed line). The data points are observational results at [FORMULA] as in Fig. 11.

However, the quantitative difference with the case [FORMULA] for the bright end of the luminosity function is quite dramatic, and thus appears to be extremely sensitive on the cosmological parameter [FORMULA].

Fig. 24 shows the B counts for the PS and non-linear scaling prescriptions, as well as for the no-evolution model. As was the case for [FORMULA], the slope of the counts gets smaller at faint magnitudes [FORMULA].

[FIGURE] Fig. 24. The B band differential number counts. The graph shows [FORMULA] (number of galaxies per square degree per apparent magnitude) for the non-linear scaling approach (solid line), the PS prescription (small dashed line), and a no-evolution model (dot-dashed line) such that the comoving luminosity function does not change with z. The data points are as in Fig. 21.

Fig. 25 shows the redshift distribution of galaxies selected at the apparent magnitude [FORMULA]. Once again we can see that the evolution is slower than for [FORMULA]: the peak moved to higher redshifts [FORMULA].

[FIGURE] Fig. 25. Redshift distribution of galaxies with apparent B band magnitude [FORMULA]. The graph shows [FORMULA] for the non-linear scaling approach (solid line), the PS prescription (small dashed line), and the no-evolution model (dot-dashed line). The data points are from Cowie et al. (1996) for galaxies in the range [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999
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