In the standard cosmological model gravitational structures in the universe have formed by the growth of small density fluctuations present in the early universe (Peebles 1980). These perturbations may be due to quantum mechanical effects and are likely to be gaussian, so that they are described by their power-spectrum. In many cases, like the CDM model (Peebles 1982; Davis et al.1985), the power increases at small scales which leads to a hierarchical scenario of structure formation. Small scales collapse first, building small virialized objects which merge later to form broader and broader halos as larger scales become non-linear. These halos will produce galaxies or clusters of galaxies, according to the scale and other physical constraints like cooling processes. Hence it is of great interest to understand the evolution with time of the density field and the mass functions of various astrophysical objects it implies. This is a necessary step in order to model for instance the formation of galaxies or clusters, which can later put constraints on the cosmological parameters using the observed luminosity function of galaxies or the QSO absorption lines.
Within this hierarchical framework an analytical model for the mass function of collapsed (or just-virialized) objects was proposed by Press & Schechter (1974) (hereafter PS). Numerical simulations (Efstathiou et al.1988; Kauffmann & White 1993) have shown this mass function is similar to the numerical results, although the agreement is improved by a modification of the usual density threshold used in this model (Lacey & Cole 1994). However, this pres cription encounters some serious defects, like the cloud-in-cloud problem studied by Bond et al.(1991), and a well-known normalization problem since one only counts half of the mass of the universe. Moreover, underdense regions are not modelled by this approach, and are simply taken into account in fine by a global multiplication of the mass function by a factor 2 which allows to get the correct normalization (this multiplicative factor 2 was recovered more rigorously by Bond et al.1991 for a top-hat in k, but this does not extend to more realistic filters). Finally, this model only predicts the mass function of collapsed objects, while one may also be interested in mass condensations of various density, in order to study galaxies or even underdense structures ("voids") for instance.
A method to describe the density field itself, and then derive the multiplicity functions of any objects, is to consider the counts-in-cells. These are closely related to the many-body correlation functions and were studied in detail by Balian & Schaeffer (1989) in the highly non-linear regime within the framework of the stable clustering picture (Peebles 1980), where the correlation functions are scale-invariant. This assumption and the scaling it implies for the matter distribution are indeed verified by numerical simulations (Bouchet et al.1991; Colombi et al.1995) and observations (Maurogordato et al.1992). Bernardeau & Schaeffer (1991) and Valageas & Schaeffer (1997) (hereafter VS) described some of the consequences of this model for the multiplicity functions of various objects like clusters or galaxies. Indeed, a great advantage of this approach is that one can study many different astrophysical objects (and even "voids") from a unique model which is not restricted to just-collapsed objects. Moreover, it is based on a rather general assumption, which does not depend on the exact details of the dynamics, and VS showed that the PS prescription can be recovered as a particular case among the possible models this scale-invariant approach can describe. Note that since the PS approach is unlikely to give a very precise description of the clustering process, because of its simplicity and the problems it encounters, the scale-invariant method offers the advantage to provide a simple and natural way to take into account the possible corrections to the PS prescription.
In this article, we intend to show how a simple PS-like model, based on the spherical dynamics, can describe the evolution of the density field and provide a specific model for the scale-invariant picture studied in Balian & Schaeffer (1989) or VS. Thus, we consider both the quasi-linear () and highly non-linear () regimes (where as usual is the amplitude of the density fluctuations at scale M given by the linear theory at the considered time), to relate the final non-linear density field to the initial conditions and to show how the scale-invariance of the many-body correlation functions and the scalings of the multiplicity functions described in VS can arise from the hierarchical structure formation scenario. We focus on the counts-in-cells, which provide a powerfull description of the density field and can be used to obtain any mass function of interest as detailed in VS. First, we derive the statistics of the counts-in-cells which our spherical model implies in the quasi-linear regime (Sect. 2). We show in particular that we recover in the limit the whole series of the cumulants derived rigorously by Bernardeau (1994a) from the exact equations of motion. We also consider the predictions of this model for the statistics of the divergence of the velocity field (Sect. 3) and explain why this simple spherical dynamics works so well in the limit (Sect. 4). Finally, we study the non-linear regime (Sect. 5). In addition to the virialization process of overdensities we take care explicitely of the evolution of underdensities. This new prescription solves in a natural way the normalization problem (in the sense that all the mass will eventually get embedded within overdense halos) and is compared with a 1-dimensional adhesion model. It also allows to complete the comparison with the scale-invariant approach. Indeed, although we do not obtain as detailed results for the mass function as the PS approach, which we believe anyway to be rather illusory since such simple and crude descriptions cannot provide rigorous and exact predictions, we think our model allows to understand how scaling properties appear, and it can predict asymptotic behaviours like the slope of the power-law tail of the mass function or its exponential cutoff (so that the model can be tested). Moreover, it gives a usefull reference which would enable one to evaluate the magnitude of the effects neglected here from a comparison with the results obtained by a more rigorous calculation.
© European Southern Observatory (ESO) 1998
Online publication: August 27, 1998