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

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7. Conclusion

In this article we build a model of galaxy formation and evolution based on the hierarchical clustering picture.

The counts of non-linear objects are done directly at the epoch under consideration using the actual non-linear density field (instead of its linear extrapolation) by methods (Schaeffer 1984; Balian & Schaeffer 1989) that have been shown to represent accurately the density field obtained in numerical simulations (Bouchet et al. 1991; Colombi et al. 1997). The result, discussed in detail in a previous paper (VS), bears some qualitative resemblance with the Press-Schechter prescription (which explains why the latter appears to be accurate in some cases), but differs quantitatively. Indeed, the Press-Schechter approximation predicts fewer extreme objects (i.e. with a density contrast much larger or smaller than the average fluctuation at fixed mass, or with a mass much smaller or larger than [FORMULA] at fixed density threshold [FORMULA] where [FORMULA]) and too many intermediate halos (VS). Moreover, our approach allows us to go beyond the traditional Press-Schechter prescription . Thus we consider mass functions of objects which are defined by a density contrast [FORMULA] which depends on their mass M (see also Valageas et al. 1999a).

Indeed we do not assume that galaxies are defined by a constant density threshold resulting from the virialization condition, since with such an assumption one would count induely virialized clusters as galaxies (Schaeffer & Silk 1985). We add to the virialization condition a cooling constraint (Silk 1977; Rees & Ostriker 1977) which separates galaxies and clusters. It states that in order to form a galaxy a non-linear object must see its baryonic content cool and settle in its central part before it gets embedded into a larger non-linear halo . More precisely, galaxies are the largest patch which satisfies these criteria. Virialized objects which do not fulfill this constraint become groups or clusters of galaxies. Note moreover that these groups are divided into several distinct galaxies which we take into account as individual galactic halos (which satisfy the cooling condition). Then, we can see that in the present universe small and faint galaxies are still defined by the virialization condition, which is more restrictive than the cooling constraint for these small virial temperatures, and are thus the result of many mergers which have kept going on until the present epoch. On the contrary, massive [FORMULA] galaxies are given by the cooling constraint, which shows that the merging process is no longer effective and their evolution is determined by internal physical processes. At earlier epochs (higher redshifts), the proportion of objects which follow the virialization condition, and are thus the result of continuous merging processes, gets larger. The [FORMULA] galaxies reach this regime at [FORMULA], a value which is determined solely by the density threshold corresponding to virialization ([FORMULA]) and the cooling processes. This epoch represents the transition from a regime of continuous mergers to a regime where galaxies evolve quietly and can form a disk.

To obtain the luminous properties of galaxies we add to this evolution of the number of galactic halos a model of star formation. We use a star formation time-scale proportional to the dynamical time (which is also the period of revolution of a proto-stellar cloud around the galaxy). In addition supernovae winds are assumed to slow down the star formation rate as they can eject or heat some gas (White & Rees 1978; Dekel & Silk 1986; White & Frenk 1991). To this purpose we construct a two-components model where the gas is expelled to the outskirts of the dark halo with an efficiency that is large when the potential well is shallow (which corresponds to faint galaxies) and is not available for star formation until it eventually falls back into the baryonic core. This ejected gas may be the source of the metal-rich intergalactic gas in clusters, which we shall study in a forthcoming article (Valageas & Schaeffer 1998). All unavoidable parameters associated with star formation processes are adjusted to the well-known properties of the Milky Way (mainly its luminosity and the gas fraction in the solar neighbourhood), and not to the luminosity function to be reproduced.

Then, our model reproduces the observed Tully-Fisher relation, the characteristic luminosities, masses and radii of galaxies . Linked with our prescription for the galaxy mass function, it provides a prediction for the galaxy luminosity function which is consistent with observations for a critical universe [FORMULA] and a low density universe [FORMULA] with a CDM (or [FORMULA]) initial power-spectrum. However, in this latter case the lack of computational results to which our non-linear scaling model could be compared makes our prediction less certain. On the other hand, this very model with the same initial power-spectrum and star formation processes shows important discrepancies with the observed luminosity function when we use a Press-Schechter-like mass function. This discrepancy cannot be healed without giving away the agreement with the above observations , and was well-known in this type of approach (inconsistency with the Tully-Fisher relation, too many small galaxies, incorrect color evolution with mass). This, obviously, is no longer a problem in our approach, the improvement arising from our better understanding of the mass function and our more logical assumption that baryon concentrations separate when they are too large for cooling to occur within a Hubble time at formation.

One interesting new feature which arises from this quite classical model of star formation, but less classical with respect to the mass function and the definition of galaxies, is that massive galaxies are more evolved and redder than small ones , as is observed, although our clustering picture is entirely hierarchical. This was not the case for earlier hierarchical models which often used a constant density threshold to define objects. For this reason, it is often stated that hierarchical models do not lead to the correct colors of galaxies and have difficulties to produce very bright and red galaxies. This statement is shown here not to be related to the hierarchical clustering picture, but to the modelling of galaxies themselves. Note that dust extinction and metallicity effects (not included in this article) would also make bright galaxies redder than would be obtained otherwise (Kauffmann & Charlot 1998; Somerville & Primack 1998). The metallicity produced by our star formation process is consistent with the observed correlation with luminosity. It is interesting to note that for the Milky Way the metallicity of stars formed during the early merging regime at [FORMULA] is [FORMULA] which is indeed the metallicity cut between disk and halo stars.

Finally, our analytic model also enables us to derive explicit approximate scaling relations between various properties of galaxies. This clearly shows the consequences of the phenomenological prescriptions adopted for star formation or feed-back processes for instance, as well as the variation of galactic characteristics with the cosmological parameters. In addition, we recall here that our model is part of a global description of structure formation in the universe, which deals with clusters, Lyman-[FORMULA] clouds and reionization processes (detailed in other articles) as well as with the galaxies we studied in this article. Thus, our goal is to provide a simple realistic analytic model which can describe in a unified consistent fashion these various phenomena. Despite the simplifications envolved in such an approach, this allows one to obtain reasonable quantitative estimates of these physical processes, to draw an explicit link between these different aspects of the same underlying density field and to get a complementary tool to numerical calculations.

Our approach differs from earlier works by many aspects. White & Frenk (1991) use the Press-Schechter approximation to count overdensities with a fixed density threshold, and within each of these halos consider the cooling condition to determine the fraction of gas that is able to cool, arguing the latter will eventually turn into stars. Cooling, in this approach, does not modify the number of objects, but simply the properties of the latter. Thus they do not solve the old problem (Schaeffer & Silk 1985; White et al. 1987) of predicting too many low-mass galaxies and extremely massive large galaxies (more precisely, they only obtain a "halo luminosity function" where the most massive objects are not galaxies but groups or clusters). For similar reasons, in order to avoid the formation of too many huge galaxies Kauffmann et al. (1998) and Somerville & Primack (1998) impose an ad-hoc cutoff ([FORMULA] km/s or [FORMULA] km/s) to the halos which are allowed to cool. This problem is cured in our model by the separation of galaxies and galaxy clusters by means of a mass-dependent density threshold as implied by our new implementation of the cooling constraint. The higher density threshold insures the proper falloff at a few [FORMULA] and the slope around [FORMULA] K of the cooling curve (see Fig. 2) provides for the flat slope of the luminosity function. We still get a rather steep slope for very small luminosities, similar to the one observed (see comments by Driver & Phillipps 1996).

Blanchard et al. (1992) discuss the problem of overpredicting the small objects, and mention the Press-Schechter prescription might not be the exact answer (but give no specific alternate prescription), blaming thus the estimate of the mass function. However, the same problems appear in numerical simulations (White et al. 1987). In fact, we find that although the estimate of the mass function by the non-linear scaling formulation does reduce the number of faint galaxies, our solution is mainly in implementing the proper effects of cooling. Blanchard et al. (1992) on the other hand suggest another way out, by reheating the universe so as to destroy the objects with the shallower potential. With their solution, the smaller the object the more efficient the destruction mechanism, and the slope of the galaxy luminosity function may be expected to be flat down to the faintest galaxies, whereas for our solution the slope eventually gets steep ([FORMULA]) for the very faint objects.

Kauffmann et al. (1993) follow White & Frenk (1991) but focus their attention on the individual histories of mass condensations. They note that a fixed density threshold implies that most massive galaxies generally formed recently and consequently have young stellar populations and blue colors, at variance with what is observed. With our views, in total agreement with the assumption of hierarchical clustering, the more massive objects have a higher dark matter density contrast (which does not imply that their density within their luminous radius is larger than for faint galaxies) and formed at high redshift. Since then, they have formed stars as almost isolated systems and have been able to exhaust their gas, whence exhibiting older stellar populations. In fact, as far as star formation is concerned, massive galaxies have roughly the same age as other galaxies, that is about the age of the universe, since before they reached the curve [FORMULA] gas was already being transformed into stars, within sub-units which later merged to create the large galaxies we can observe today. In this sense, the "age" of these bright galaxies (that is the time available for star formation) is even slightly larger than the one of small galaxies, because within our model these massive objects are the final results of the merging of deep potential wells where star formation started earlier since cooling was more efficient there than in weaker halos. However, the main reason in our model (with the fact that all galaxies have roughly the same "age") for the old stellar population of luminous galaxies (as compared to faint ones) is that their star formation process is more efficient because of their high density and virial temperature. This trend is also apparent (but to a lesser degree) in the model of Cole et al. (1994).

The study of the galaxy counts as a function of apparent magnitude, where evolution in luminosity and in number are intrinsically tied (Guiderdoni & Rocca-Volmerange 1990) is not fully satisfactory within our simple description. It deserves a detailed study using our methods but including properly color evolution, and remains to be done with the needed care. Also, the fate of galaxies in the dense parts of clusters, when dark halos start to loose their individuality, would be worth a thorough consideration. Clearly, however, the obvious outcome of such a model where mass and luminosity evolution are treated consistently is the study of the galaxy populations at high redshift. We have here only unveiled some of the applications of our approach.

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© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999