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

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1. Introduction

The formation and evolution of galaxies can be studied through numerical simulations or analytic (or semi-analytic) models. In principle, the former could follow with great details the numerous processes (growth of density fluctuations, galaxy interactions, cooling processes, star formation histories, effects of supernovae and metallicity...) which govern galaxy formation. However numerical simulations are much less flexible than analytic means, and they also have to use physical parameterisations to model key processes like star formation, not only because the required dynamic range would exceed current computational possibilities but also because these physical processes are not yet very well understood. Since the final results depend strongly on such parameterisations (Navarro & White 1993), analytic or semi-analytic models provide an attractive alternative way to study galaxy formation, as they clearly show the relative importance of different processes and the influence of the available parameters, which would be more difficult to understand from a complex N-body and hydrodynamic simulation.

The standard calculations follow the ideas of Press & Schechter (1974) (see also Schaeffer & Silk 1985), taking advantage of the progress made originally by Bond et al. (1991), currently referred to as the "excursion set formalism". The results obtained by such models (White & Frenk 1991; Kauffmann et al. 1993; Cole et al. 1994; and subsequent articles) show a reasonably good agreement with various observations which suggests that the main features of the standard scenario (galaxies arise from gas which is able to cool within dark matter halos formed by a hierarchical clustering process) are correct (or provide at least a good approximation). However, several problems are still unsolved: these studies cannot recover simultaneously the normalization of the Tully-Fisher relation and of the B-band luminosity function, the slope of the luminosity function at the faint end is too steep, massive and very bright galaxies usually are not redder than fainter ones contrary to observations and the models often predict a cutoff at the bright end which is too smooth (there are too many very luminous objects). We present in this article an analytic model for galaxy formation and evolution, based on a specific description of density fluctuations in the highly non-linear regime and an original application of the cooling constraints, which provides a solution to these issues and allows detailed predictions for many physical galactic properties together with their redshift evolution.

Thus, the motivations of this study are to:

  • take advantage of an improvement (Valageas & Schaeffer 1997, hereafter VS) over the usual Press-Schechter mass function (Press & Schechter 1974), that originates in the understanding of the density fluctuations in the deeply non-linear regime (Schaeffer 1984; Balian & Schaeffer 1989; Bernardeau & Schaeffer 1991) and amounts to include the important subsequent non-linear evolution by counting directly the overdensities at the epoch of interest rather than in the primordial universe as implied by the Press-Schechter approximation. Since in this approach objects are defined by taking a snapshot of the actual density field, in order to get the mass function of dark matter halos there is no need to follow the merging history of the various objects . This is done implicitely since the time evolution of the density field is already built-in within this non-linear hierarchical scaling model (and well verified by numerical simulations, see Bouchet et al. 1991; Colombi et al. 1997; Munshi et al. 1998; Valageas et al. 1999a). Note however that the merging history of the dark matter halos enters, but only marginally, when the star formation history is considered. This is discussed in Appendix C.

  • implement in a natural way the cooling constraints . Indeed, the difference between galaxies and groups (that may have similar masses!) is that the former are able to cool rapidly (Rees & Ostriker 1977; Silk 1977; White & Rees 1978) whereas the latter are not. We shall argue that for a given velocity dispersion, galaxies are made from the largest possible baryonic patch that can cool within a few Hubble times at formation . By contrast, a larger patch which cannot cool rapidly in a uniform fashion (due to its lower overall density which translates into a larger cooling time) will become a group or a cluster of galaxies while several high density sub-units will cool and form distinct galaxies. Indeed, the non-linear scaling model predicts strong sub-clustering so that large clusters automatically contain numerous very high density spots that with our method we count as galaxies. For a galactic mass M our constraints translate into a well-defined mass-dependent condition on the galaxy radius [FORMULA], or equivalently on its density contrast [FORMULA], rather than having a universal (i.e. mass-independent) density contrast for all objects. Our approach is, in this respect, fundamentally different from the earlier ones .

    We apply this formulation to two descriptions of gravitational clustering, based i) on a Press-Schechter-like approach that uses linear theory and ii) on our non-linear hierarchical scaling model (VS). We show that in both cases it leads to a strong bright-end cutoff as required. As a result, we

  • show how the previous puzzles (normalizations of the Tully-Fisher relation and B-band luminosity function, flat slope for the latter, colors) can be solved within a hierarchical scenario. We can note that although relating the luminosity to the mass of the galaxy implies a model of star formation with a few adjustable parameters, observational constraints (Tully-Fischer relation, gas/stars mass ratio,...) allow one to derive a mass-luminosity relation for galaxies independently of any consideration of the galaxy luminosity function. Hence the agreement with the observed luminosity function builds confidence in our model.

  • obtain detailed predictions for the redshift evolution of galaxies. Indeed, since there is nothing special about the present epoch the same considerations predict the evolution of the galaxy mass and luminosity functions in the past (as well as in the future!).

  • take advantage of the possibilities of our analytic model to get scaling relations , analogous to the Tully-Fisher relation, between various physical quantities (circular velocity - luminosity - mass - gas/star mass ratio - metallicity - star formation rate) over well-defined ranges of galaxy luminosity (for instance). Obtaining such laws, which requires the use of an analytic approach, provides a deep level of understanding and a clear presentation of the global trends implied by models based on the standard hierarchical scenario, independently of the details of both the star formation process and gravitational clustering. We think this transparancy and flexibility makes it worthwile to develop analytic models like ours (despite the simplifications they involve) in addition to numerical approaches.

  • present a model for galaxy formation within a global description of gravitational structures and astrophysical objects which provides a unified consistent model for very different phenomena: from clusters (Valageas & Schaeffer 1998) to Lyman-[FORMULA] clouds (Valageas et al. 1998) and galaxies.

The organization of the paper is as follows: in Sect. 2 we discuss the determination of the mass function, with two models, one, that we call PS approach , based on an educated guess of which fluctuations identified in the early universe are going to eventually become galaxies, as prescribed by Press & Schechter (1974) but with a mass-dependent density contrast, and a second one, based on our previous work, using the same considerations applied directly to the non-linear density field at the epoch under consideration, that we call non-linear hierarchical scaling approach . In Sect. 3 we discuss the cooling condition, as well as our star formation model. In Sect. 4 we construct the galaxy luminosity function at the present epoch using the PS model as well as the non-linear scaling approach for a critical and an open universe. The evolution effects are examined in detail in Sect. 5. Finally, in Sect. 6, these considerations about the mass function and its time evolution are applied to derive the quasar distribution as a function of redshift, with an improvement over the Efstathiou & Rees (1988) model due to our new mass function that replaces the original one derived using the Press-Schechter model. In the last section we summarize the results of this paper and discuss its similarities and differences with other papers on the same subject (White & Frenk 1991; Blanchard et al. 1992; Kauffmann et al. 1993,1998; Cole et al. 1994).

The details of our model are presented in Appendix. In particular, we describe our star formation model in Appendix B and Appendix C. We discuss the values of our parameters in Appendix F and we present scaling relations in Appendix G. We use [FORMULA] km/s/Mpc throughout this paper.

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Online publication: April 19, 1999