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

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4. Numerical results

Now we can compute numerically the galaxy luminosity function, as defined in the previous section, as well as the physical characteristics of the halos we consider.

4.1. [FORMULA]

4.1.1. Halo properties

In the case [FORMULA], Fig. 3 shows the relation temperature-density which corresponds to the cooling and virialization constraints (Fig. 2), at redshift [FORMULA]. We see that the typical mass and radius of the halos we get are close to the observed values for galaxies in the present universe. For [FORMULA]K the cooling constraint translates into a nearly constant mass [FORMULA], while at large masses, or large temperatures [FORMULA]K, it translates into a fixed radius [FORMULA]kpc. Indeed, the curve [FORMULA], see (17), can also be written:

[EQUATION]

At high temperatures, we can write [FORMULA] with [FORMULA] K and [FORMULA]. This gives a constant radius R:

[EQUATION]

[FIGURE] Fig. 3. Temperature-density diagram given by the cooling and virialization constraints in the case [FORMULA]. The short-dashed lines correspond to halos of fixed mass, on the left, and halos of fixed radius, on the right. We note [FORMULA] the critical density at [FORMULA].

The fact that we know the radius of the halos we consider to be at least 60 kpc (because a few rotation curves measured in large spiral galaxies remain flat at least up to this radius) requires that [FORMULA]. However, as we explain in Appendix F the product [FORMULA] is determined independently by the luminosity of the Milky Way. This leads to [FORMULA] in the case [FORMULA]. Hence both constraints can be satisfied simultaneously. Since we use [FORMULA] as given by the simple spherical collapse picture and we expect [FORMULA] and we noticed in Sect. 3.1 that [FORMULA], we choose [FORMULA] and [FORMULA]. This gives [FORMULA] kpc in the limit of very large virial temperatures.

Galaxies with small virial temperature [FORMULA] K and rotation velocity [FORMULA] km/s are irregular or (dwarf-) elliptic galaxies, dominated by their continuous merging history, while massive galaxies with [FORMULA] km/s are disk galaxies which have evolved through a much calmer history since they reached the curve [FORMULA]. Indeed, they remain unchanged, or slowly accrete some mass if clustering is not exactly stable, so that a disk can form. This boundary [FORMULA] km/s (which corresponds to a B-band magnitude [FORMULA]) is indeed close to the observed transition between spirals and (dwarf-) ellipticals or irregulars (Sandage et al. 1985).

We can see that in the near future ([FORMULA]) two halos of the same mass or the same radius can have different temperatures and densities. This implies that, within the framework of this model, neither the mass nor the radius are good variables to describe the halos we consider, while the velocity dispersion (or the temperature) is. Note that the latter is very close to our variable x that we think is the true parameter characteristic of mass condensates.

We notice that light and faint galaxies, which correspond to small temperatures or velocity dispersions, are young objects characterized by a density contrast [FORMULA]. Indeed, these small galaxies are on the curve [FORMULA] in the cooling diagram (see Fig. 3). This may look surprising, since the observed density contrasts of these objects are usually very large. However, this is simply explained by the fact that the observed effective luminous radius [FORMULA] is much smaller than the radius we consider which describes the actual size of the dark matter halo from where the gas content of the galaxy originated. This large difference in radius translates into the density contrast we obtain. The small size of the observed luminous radius of these objects, that is not modelled here, can be explained by several effects. For instance a large fraction of gas is ejected, since the potential well is too weak to retain baryons as efficiently as in large galaxies. Also star formation is not very efficient so that it cannot take place very far from the center of the galaxy because the density gets quickly too small.

We do not attempt to model here the radius and the mass within the luminous part. This has already been done in a similar context (Bernardeau & Schaeffer 1991). It was seen there by phenomenologically modelling the baryon squeezing due to cooling, that the observed behaviour of the luminous radius calls for a dark matter halo radius that behaves nearly as [FORMULA] for the smaller masses (i.e. a constant density contrast) but for an almost constant dark halo radius for the large masses (due to a change in the [FORMULA] ratio in these two regimes). These findings, indeed, encouraged us to undertake the present work. For instance, if [FORMULA] and the density profile of galactic halos is a power-law [FORMULA] ([FORMULA]), then for small galaxies on [FORMULA] where [FORMULA] the density contrast within the luminous radius scales as [FORMULA] while for bright galaxies on [FORMULA] which have a nearly constant dark matter radius we get [FORMULA]. Hence the density within the luminous radius decreases for brighter galaxies even though the density within the much larger dark matter halo increases. However, a precise estimate of these relations would require a detailed model of the behaviour of the gas after cooling which we do not consider in this article.

4.1.2. Gas/star mass ratio

The variation of the gas/star mass ratio, which is closely related to the ratio (star formation time-scale)/(age of the galaxy), see (D4), is shown in Fig. 4 as a function of the galaxy circular velocity. Small [FORMULA] galaxies which have a small star formation rate [FORMULA], because of their low density and virial temperature, have a high gas/star mass ratio [FORMULA]. On the other hand, high temperatures at the external radius of the halo, or large [FORMULA], correspond to high densities. Hence these galaxies have undergone a very efficient star formation, since their star formation rate [FORMULA] is large. As a consequence, they have already lost most of their gas content, which turned into stars, as we can see in Fig, 4, and [FORMULA]. As expected on general grounds, see (D5), we verify that [FORMULA], corresponding to a galaxy similar to the Milky Way, implies [FORMULA] (we have [FORMULA] and [FORMULA] since there is some gas in the halo: [FORMULA]). This value of the gas/star mass ratio leads us to choose for our last parameters [FORMULA], [FORMULA] and [FORMULA] K. We can see in Fig. 4 that while most of the gas is in the dense component [FORMULA] for galaxies brighter than the Milky Way (with a larger rotation velocity), it is mostly in the diffuse phase [FORMULA] for faint galaxies, as we can see in (B5), because supernovae are very efficient in these weak gravitational potentials and eject the gas out of the star-forming regions. In particular, from (B5), (B9) and (B10) we see that for faint galaxies [FORMULA] while [FORMULA] for bright galaxies.

[FIGURE] Fig. 4. Ratios of the total mass of gas [FORMULA] (solid line) and of the mass of star-forming gas [FORMULA] (dashed line) over the mass [FORMULA] of luminous stars as a function of the circular velocity [FORMULA].

One may distinguish three different regimes (Appendix G) in which approximate power-law relations can be derived.

1) Very faint galaxies, located on [FORMULA] and [FORMULA] K, with a constant density contrast .

2) Faint galaxies, located on [FORMULA]: [FORMULA] kpc and [FORMULA] K, with a nearly constant radius .

3) Bright galaxies located on [FORMULA] kpc and [FORMULA] K, which have nearly exhausted all their gas .

Using these analytic estimates (Appendix G) it is readily seen that the gas/star mass ratio is a steep function of the circular velocity ([FORMULA] to [FORMULA] and finally it follows an exponential decline). We can see very distinctly in Fig. 4 these three regimes.

It is important to realize that these relations should be approximatively valid for any model of star formation. The relation (D5) is always correct for [FORMULA], where [FORMULA] is the star formation time-scale. Indeed, for [FORMULA] we have [FORMULA] by definition of [FORMULA], hence [FORMULA]. Since the Milky Way is characterized by [FORMULA], fainter galaxies (which have a higher gas/star mass ratio) verify [FORMULA], [FORMULA] and [FORMULA], while more luminous galaxies satisfy [FORMULA] and [FORMULA]. This means that the relation [FORMULA] for these bright galaxies, if the stellar mass/luminosity ratio [FORMULA] is roughly constant, is directly given by the physical characteristics of the underlying halos - since [FORMULA] - hence by the cooling curve, and not by the details of star formation processes. Hence the slope of the corresponding Tully-Fisher relation (which is a bit shallow) could only be modified by a change of the definition of galactic halos (i.e. using another constraint than the cooling criterion) or by a gas/dark matter mass ratio which would vary with the galactic characteristics (one may argue that deep potential wells could gain some gas from surrounding small halos). On the contrary, the luminosity of faint galaxies [FORMULA] depends strongly on the star formation rate. Using [FORMULA] and [FORMULA] (where [FORMULA] is the age of the universe), we have [FORMULA]. Hence the dependence of [FORMULA] on V is constrained by the observed Tully-Fisher relation. In fact in our model the relation we obtain in this case is [FORMULA] but it leads to results which are still consistent with observations. Thus, the high uncertainty which lies in any star formation prescription (since this process is rather poorly known) is greatly reduced by these general properties and the observed Tully-Fisher relation. It is also apparent through these considerations (and is confirmed by a simple calculation) that the use of a constant star formation time-scale ([FORMULA] for all times) would not change these results (at [FORMULA]), except for small details. As a consequence, we can reasonably expect our results to be fairly general and robust.

4.1.3. Luminosity

Galaxies characterized by high temperatures and large [FORMULA] also correspond to the most luminous galaxies, and their luminosity is mainly due to small stars which have a long life-time, since there is not much gas left to create new generations of massive short-lived stars, as it appears in Fig. 5. We can see in the lower panel that the Tully-Fisher relation is approximately satisfied, although the slope of the relation [FORMULA] gets shallower at high luminosities. For B-band luminosities, observations give [FORMULA] (Kraan-Korteweg et al. 1988; Pierce & Tully 1988), while the infrared relation is steeper: [FORMULA] (Pierce & Tully 1992). In our model, these relations are obtained (Appendix G) as a smooth transition from [FORMULA] at the very faint end to [FORMULA] at the bright end.

[FIGURE] Fig. 5. Upper figure: ratio of the total mass of luminous stars [FORMULA] over the galaxy luminosity L. Lower figure: luminosity L of the galaxy as a function of [FORMULA]. The thin dotted line corresponds to the power-law [FORMULA] normalized to the Milky Way. The short-dashed curve is the luminosity due to long-lived small stars ([FORMULA]) while the dot-dashed curve is the luminosity due to short-lived massive stars ([FORMULA]).

Hence the relation [FORMULA] we obtain has a strong slope close to the usual Tully-Fisher relation, and gets shallower for galaxies more luminous than the Milky Way. It is interesting to note that Persic & Salucci (1991) found a similar decrease of the slope of the Tully-Fisher relation for luminous galaxies in observations, although in the infrared H-band. We can note in Fig. 5 that for faint galaxies the slope is close to 3 and not 5 as in the approximation obtained in Appendix G because the ratio [FORMULA] is not exactly constant and increases for large circular velocities.

As we can see from (D1) and (D4), this variation of the mass/luminosity ratio [FORMULA] is due to the fact that large and bright galaxies (also characterized by a large temperature or circular velocity), which have a high star formation rate [FORMULA], have already consumed most of their initial gas content. Hence, their present star formation rate is relatively small (compared to their past history), and their stellar population presents an increasing proportion of old stars, with a long life-time, which were created during the whole life of the galaxy. Thus, as we can see in Fig. 5 the luminosity in the form of short-lived massive stars is much smaller than the contribution of long-lived small stars. On the contrary, small galaxies which still have a relatively high star formation rate (as compared to their past) because of their large gas content show an important contribution from massive stars which are created at the current epoch. In fact, both classes of stars give roughly the same luminosity. This is due to the fact that the luminosity of the galaxy is dominated by stars of intermediate mass [FORMULA], as described in Appendix D. This variation of the mass/luminosity ratio shows that very massive galaxies should be redder than small ones, which is consistent with observations (Lilly et al. 1991; Metcalfe et al. 1991). Moreover, this would be enhanced by metallicity effects, since bright galaxies are also the most metal-rich, which we recover in our model as we shall see now.

4.1.4. Metallicity

Next, we can also consider the metallicities [FORMULA] (diffuse gas), [FORMULA] (dense gas) and [FORMULA] (stars), which we introduce in Appendix E. They are displayed in Fig. 6, as a function of the B-band luminosity of the galaxy. For faint galaxies we are in the "stationary" regime where [FORMULA] and [FORMULA]. The variation of the metallicities [FORMULA] and [FORMULA], which are those available to observations, agrees with the data. We chose the value of the yield y so that we get [FORMULA] for the Milky Way. This gives [FORMULA].

[FIGURE] Fig. 6. Metallicity Z as a function of the B-band magnitude [FORMULA]. The solid line corresponds to the gas in the core ([FORMULA]), the short-dashed line to stellar metallicities ([FORMULA]) and the dot-dashed line to the gas in the halo ([FORMULA]). The data points are observations from Zaritsky et al. (1994).

As usual, we can consider the 3 regimes we introduced previously to determine approximate relations for the metallicity (Appendix G). For galaxies fainter than the Milky Way we have [FORMULA], or [FORMULA]. Galaxies more luminous than the Milky Way have a constant metallicity [FORMULA] while [FORMULA]. Note however that the yield may vary with the characteristics of the galaxy, together with the IMF. Nevertheless, we see that our model agrees with observations over 10 magnitudes in [FORMULA].

4.1.5. Stellar history

Finally, we can consider (rather crudely) the mean morphological properties of the galaxies we obtain in our model as a function of their B band magnitude [FORMULA]. We define an approximate disk/bulge luminosity ratio by:

[EQUATION]

when [FORMULA] ([FORMULA] is the Hubble time at the considered redshift and [FORMULA] the virialization time of the galaxy). Thus, we evaluate the disk luminosity as the contribution from stars formed after the galaxy reached the cooling curve [FORMULA], and the bulge luminosity as the contribution from earlier stars. For galaxies which have not reached [FORMULA] yet we use [FORMULA] since there is no disk. This approximation is only very crude, as the disk may not form exactly at the virialization time [FORMULA], and some stars probably form in the bulge after this date. The variation of this ratio [FORMULA] with [FORMULA] is shown in Fig. 7.

[FIGURE] Fig. 7. The variation of the disk/bulge luminosity ratio [FORMULA] with the B band magnitude [FORMULA].

Faint galaxies [FORMULA] have not reached [FORMULA] yet, hence they have no disk and are elliptical or irregular galaxies. For [FORMULA] the disk/bulge luminous ratio first increases with luminosity, since brighter galaxies (which also have a higher density, virial temperature and a larger mass) have been on [FORMULA] since a higher redshift. However, this ratio declines for very luminous galaxies [FORMULA], which correspond to large densities and deep potential wells, hence to very efficient star formation. As a consequence, these galaxies transformed a large part of their initial gas content into stars at very high redshift, while they were still divided into several sub-units, before they reached the curve [FORMULA] which marked the end of their merging phase of formation. Hence, the mass of stars formed since this latter epoch is increasingly small as compared to the stellar population formed during the merging phase, as the galaxy parameter x increases. Thus, it appears naturally in our model that spiral galaxies only correspond to an intermediate range of luminosities, [FORMULA], while brighter and fainter galaxies should be ellipticals or irregulars. Hence we get old bright ellipticals as a straightforward outcome. This is quite similar to the observed dependence of the dominant Hubble type on the luminosity (Sandage et al. 1985). However, the interactions between galaxies in clusters would certainly add several effects which we did not take into account explicitly but could have important consequences on the mean luminosity-morphological type relation.

4.1.6. Milky Way

The Milky Way corresponds to [FORMULA] km/s, and in our model to [FORMULA] kpc, [FORMULA], [FORMULA], [FORMULA] K, [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. Its present supernovae rate is [FORMULA] which is consistent with observations: van den Bergh & Mc Clure (1994) find [FORMULA]. The above value of [FORMULA] is consistent with the values [FORMULA] of the disk surface densities measured in the solar neighborhood: [FORMULA] (Bienayme et al. 1987), 0.47 (Kuijken & Gilmore 1989), 0.5 (Gould et al. 1996). It implies a gas/(gas+stars) mass ratio of [FORMULA] which is closer to the actual observations than the value [FORMULA] traditionally taken in many calculations of the chemical evolution in the solar neighborhood. The Milky Way also corresponds to a star formation rate [FORMULA]year, which agrees with usual estimates. Note that the present ratio (stellar mass)/(age of the universe) [FORMULA] (in [FORMULA]/year) implies that any model with the correct luminosity -hence the correct stellar mass- will give a present-day star formation rate of the right magnitude.

4.1.7. [FORMULA], CDM

Now, to get the mass function or luminosity function of galaxies, we need the value of the initial power-spectrum [FORMULA], or the correlation functions [FORMULA] and [FORMULA]. We first consider the case where [FORMULA] is a CDM-like power-spectrum. More precisely, following Davis et al. (1985) we use:

[EQUATION]

where

[EQUATION]

and A is a normalization constant such that [FORMULA]. Fig. 8 shows the linear correlation function [FORMULA] and the evolved non-linear correlation function [FORMULA] at redshift [FORMULA]. The calculation of the non-linear correlation function [FORMULA] from its linear counterpart [FORMULA], using the spherical model normalized by the numerical calculation performed by Jain et al. (1995), is detailed in VS.

[FIGURE] Fig. 8. The evolved correlation function [FORMULA] at [FORMULA] (solid line), and the linear extrapolation [FORMULA] (dashed line), in the case [FORMULA] and [FORMULA] is a CDM power-spectrum. The dot-dashed line is the power-law [FORMULA] with [FORMULA]

We can see that the evolved non-linear correlation function is close to a power-law [FORMULA], with [FORMULA], in the range 50 kpc to 1 Mpc, which covers all the values that the radius of the halos we consider can take. As we saw in Sect. 2, we can now get the luminosity function of galaxies, provided we know the function [FORMULA] we introduced in that section. We choose the function [FORMULA] given by Bouchet et al. (1991):

[EQUATION]

with:

[EQUATION]

The galaxy luminosity function we get in this way is shown in Fig. 9 as a function of the B-band magnitude ([FORMULA]). The short-dashed curve is the prediction of the PS approach.

[FIGURE] Fig. 9. The galaxy luminosity function [FORMULA]. The graph shows the quantity [FORMULA] as a function of the magnitude in the B-band for the non-linear scaling approach (solid line) and the PS prescription (dashed line). The data points are observations from Loveday et al. (1992) (circles), Ellis et al. (1996) (filled squares) and Zucca et al. (1997) (triangles).

We explained in Sect. 2 that the PS approach leads to the multiplicity function [FORMULA] with:

[EQUATION]

and [FORMULA] given as a function of M by (2), the mass M being in turn related to L by the model of star formation. As we could see in Fig. 3, the typical masses of our galactic halos are [FORMULA]. As a consequence, since the cutoff of the multiplicity function given by the PS approach occurs for [FORMULA], which should correspond roughly to the cutoff of the observed luminosity function, the number density of galaxies is of the order of:

[EQUATION]

which is much larger than the value [FORMULA] given by observations. Hence the luminosity function implied by the PS approach is much larger than the observed values at luminosities smaller than its cutoff, as we can see in Fig. 9. A galaxy similar to the Milky Way, with a mass [FORMULA], a density contrast [FORMULA] and a B-band magnitude [FORMULA], corresponds to a linear parameter [FORMULA]. Hence it is already at the cutoff of the PS multiplicity function while the cutoff of the observed luminosity function corresponds rather to [FORMULA]. Thus, the luminosity function given by the PS approach falls down at luminosities smaller than what is observed, as we can see in Fig. 9. Moreover, in the case where [FORMULA] (on galactic scales [FORMULA]) we have:

[EQUATION]

This allows us to get (Appendix G) the shape of the luminosity function implied by the PS approach in the three regimes we have already considered [FORMULA], but with a rather small value for [FORMULA]. Thus, as seen in Fig. 9, the luminosity function we get in this way has a shape somewhat similar to the observations (a power-law with an exponential cutoff), but, as already discussed by VS, its normalization is too high and its cutoff is too strong.

On the other hand, the non-linear scaling approach leads to:

[EQUATION]

with x given by (8) and the function [FORMULA] by (29). Now, the cutoff at [FORMULA] must correspond to [FORMULA], so:

[EQUATION]

with a typical mass [FORMULA]. This is close to the observed values, hence the normalization of the luminosity function implied by the non-linear scaling approach is consistent with observations, as we can see in Fig. 9. We can also look at the shape of the luminosity function in the 3 regimes we have already considered. We get (Appendix G) [FORMULA] at the faint end and [FORMULA] for the bright galaxies, with a much larger value of [FORMULA] than with the PS approximation. Hence we see, Fig. 9, that the luminosity function we get in this way is quite close to observations: it shows an exponential cutoff for bright galaxies [FORMULA] and a power-law behaviour for faint galaxies [FORMULA]. The existence of a flat plateau for [FORMULA] is characteristic of our results (for both the PS and non-linear scaling approaches) and holds also for other power-spectra (e.g. [FORMULA]). This feature is quite remarkable, as it did not appear in previous models but is in good agreement with observations, which seem to show an upturn at [FORMULA] after a flat portion, see for instance Driver & Phillipps (1996). This sudden change of the slope of the luminosity function around [FORMULA] in our model corresponds to the transition from the curve [FORMULA] to [FORMULA], which define the global properties of galaxies (mass, radius,...).

As we noticed in Sect. 2 the PS mass function can be written in the same form as the non-linear scaling one, with a different scaling function [FORMULA] which has a stronger and earlier cutoff for large x and a higher normalization at [FORMULA] (see also VS). The difference between both luminosity functions we can see in Fig. 9 is a direct consequence of the difference between these scaling functions [FORMULA]. This suggests in turn that the non-linear scaling function describes the actual outcome of gravitational processes more accurately than the scaling function derived in the PS approach, and hence than the usual PS prescription. Indeed, the relation [FORMULA] is strongly constrained by the Tully-Fisher relation, which thus enables one to derive strong constraint on the scaling function [FORMULA]. Note however that there is an additional factor [FORMULA] in the luminosity functions (33), so that the power-law or the normalization of [FORMULA] cannot be constrained without a specific model for galaxies, like ours, which gives the value of M attached to the parameter x. Nevertheless, the exponential cutoff of [FORMULA] is quite strongly constrained by the one of the luminosity function. We can note that Kauffmann et al. (1998) also found that a Press-Schechter approach predicts too many intermediate galaxies as compared to the results of N-body simulations. Although their model is significantly different from ours this effect agrees with our analysis of the mass functions (see Sect. 2.3 and VS).

4.1.8. [FORMULA] and [FORMULA]

Similarly to the case of a CDM power-spectrum we can also consider the cases of a power-law [FORMULA] with [FORMULA] and [FORMULA]. The case [FORMULA] gives results very close to those obtained for a CDM power-spectrum. This is natural since the latter has a local slope [FORMULA] on galactic scales. On the contrary, the case [FORMULA] produces a very strong exponential cutoff which leads to a luminosity function quite far from the observed one. Indeed, we now have [FORMULA] instead of [FORMULA] (for [FORMULA]) which means that bright massive galaxies are much more rare relative to small ones as compared to the previous case [FORMULA], since density fluctuations decrease faster with larger mass. Thus a power-spectrum index [FORMULA] at galactic scales seems to be incompatible with the observed luminosity function, at least within the framework of our model. Moreover, as most galaxy characteristics are strongly constrained by observations (Tully-Fisher relation, gas/star mass ratio, lower limit for galactic masses and radii, ...) it is very likely that no reasonable model would reconcile the observed galaxy luminosity function with [FORMULA].

4.2. [FORMULA]

In the case [FORMULA] and [FORMULA], we choose [FORMULA] and [FORMULA]. The physical properties of the galaxies are close to those in the [FORMULA] universe, the analogs of Fig. 3, Fig. 4 and Fig. 5 show the same behaviour as in the previous case. For instance, a galaxy similar to the Milky Way corresponds to [FORMULA] kpc, [FORMULA], [FORMULA], [FORMULA] K and [FORMULA]. Note that we obtain a smaller mass and radius, as compared to the case [FORMULA], as we explain in Appendix F.

We consider the case of a CDM power-spectrum. Fig. 10 shows the linear correlation function [FORMULA] and the evolved non-linear correlation function [FORMULA] at redshift [FORMULA].

[FIGURE] Fig. 10. The evolved correlation function [FORMULA] at [FORMULA] (solid line), and the linear extrapolation [FORMULA] (dashed line), in the case [FORMULA] and [FORMULA] is a CDM power-spectrum. The dot-dashed line is the power-law [FORMULA] with [FORMULA]

We see that the evolved non-linear correlation function is still reasonably close to a power-law [FORMULA], with [FORMULA], in the range 30 kpc to 1 Mpc. The scaling function [FORMULA] we need to obtain the galaxy luminosity function in the non-linear scaling approach is not available from the current numerical simulations but as we argued in VS it is expected to be similar to the scaling function obtained in a critical universe with the same power-spectrum. Hence we adopt the function [FORMULA] we used for a critical universe, see (29). The galaxy luminosity function we get in this way is shown in Fig. 11 as a function of the B-band magnitude.

[FIGURE] Fig. 11. The galaxy luminosity function [FORMULA]. The graph shows the quantity [FORMULA] as a function of the magnitude in the B band for the non-linear scaling approach (solid line) and the PS prescription (dashed line). The data points are as in Fig. 9.

As we can see from Table F1 in Appendix F, when we shift [FORMULA] from 1 to 0.3 we should decrease [FORMULA] to keep the radius of the Milky Way larger than 60 kpc. In fact, since for a critical universe we had a large radius [FORMULA] kpc, we can keep [FORMULA] roughly constant. Thus, we choose [FORMULA] which leads to [FORMULA] kpc. From Table F1 it implies that the mass of galaxies declines roughly in the same proportion as the radius, and indeed we now have [FORMULA] for the Milky Way. The number density of galaxies given by the PS approach is now:

[EQUATION]

which is closer to observations than it was in the case [FORMULA], as we can see in Fig. 11. However, since the relation temperature - luminosity did not change (because it is constrained by the observed Tully-Fisher relation), the cutoff entailed by the PS approach is still too strong. As was the case for a critical universe, the non-linear scaling approach can produce a smoother cutoff at higher luminosities. The analysis made in Sect. 4.1.7 and Appendix F for the three regimes of galaxies still holds, hence we recover the same slopes. However, in such a low density universe the curve [FORMULA] is lower relative to [FORMULA] (as compared to a critical universe) on the analog of Fig. 3, hence even for relatively faint galaxies [FORMULA] we still are in the regime 2), thus the slope of the luminosity function is very small, for both prescriptions. In fact, as we can see in Fig. 11, the luminosity function is flat between [FORMULA] down to [FORMULA], which is quite remarkable. Overall, the agreement with observations is still very good.

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Online publication: April 19, 1999
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