Astron. Astrophys. 320, 365-377 (1997)

2. Theoretical mass function

In the gravitational instability scenario, virialized objects like galaxy clusters form from initially small density fluctuations that grow under the influence of gravity. The density field is specified by its power spectrum and the statistical nature of the fluctuations. It is generally assumed that the field is Gaussian, although some consequences of non-Gaussianity have been examined (Weinberg and Cole 1992), a question to which we return in Sect. 7. For the power spectrum, we adopt a simple power law,

over the mass range corresponding to galaxy clusters. In hierarchical models, such as CDM, the variance of the density field diverges on small scales, and so one must work with a smoothed version (see, for instance, Bardeen et al. 1986). The variance of the density field smoothed with some window function W on a scale corresponding to mass M is

where is the Fourier transform of the window function. Davis & Peebles (1983) found that the variance of galaxy counts within spheres of radius , quoted as , is close to one. If the galaxy distribution follows the mass distribution, then the variance of the density perturbations on the same scale would also be equal to unity. However, if the galaxy distribution is biased relative to the mass distribution, then the variance of mass fluctuations in a sphere of radius R containing a mass can be written in the following way:

Accordingly, in the following, the value of b will correspond to where b is the bias parameter. The value of b advocated to explain the observed abundance of galaxies, their correlations and their velocity dispersion was in the range 2-2.5. This large value of b met with difficulty in other quarters (Valls-Gabaud et al. 1989).

Since in one dimension the early stage of the non-linear collapse is entirely determined by the amplitude of the local mean density, the exact solution for the collapse of an overdensity is calculable until the first orbit crossing occurs. In three dimensions, the solution is also known for the case of a spherical matter distribution (Lemaître 1933, Gunn & Gott 1972, Peebles 1980). This so-called spherical model can be used to model the non-linear collapse of a cluster, which one then finds is driven by the value of the density field smoothed with a top-hat window of size comparable to that of the cluster. It is well known that when the linear density field reaches the value 1.68, the density becomes singular for a purely spherical collapse. In reality, the spherical symmetry is broken by the development of substructures which instead leads to the formation of a stationary state, the "virial" equilibrium. The final radius of the collapsed object is expected to be half of its maximum expansion radius, corresponding to a density contrast of the order of 200. In the absence of significant fragmentation during the collapse, initial density fluctuations of the field smoothed on the scale R are expected to collapse to structures with a typical mass , where is the mean cosmological background density. Within this framework, it is in principle possible to relate the ensemble characteristics of non-linear objects to the statistical properties of the initial density field.

In practice one deals with the initial density field linearly extrapolated to the present epoch . For instance, the rms fluctuation on some scale M is:

In this relation D is the growing mode solution of the linearized growth equation and is the redshift corresponding to some early time at which the fluctuations in the universe were still linear. For the case of and a vanishing cosmological constant, , , where a is the expansion factor.

Despite the fact that the spherical top-hat model permits a considerable simplification of the actual development of non-linearities, the precise calculation of the number density of collapsed objects of mass M remains an extremely complicated problem. One major difficulty comes from the fact that a given region of space identified as non-linear on some scale might in fact form part of a still larger non-linear structure. This is the so-called "cloud-in-cloud" problem. However, one may assume that, being rare, massive objects, such as galaxy clusters, originate from nearly isolated density fluctuations, for which the cloud-in-cloud effect should be less important. In this case, the spherical model is likely to be a good description of the nonlinear evolution. Indeed, Bernardeau (1994) has shown that the rare density fluctuations of a Gaussian random field follow exactly the dynamics of the spherical model.

Using the spherical model, Press & Schechter (1974) proposed a derivation of the mass function of virialized objects. They argued that the fraction of matter in the form of non-linear objects with mass greater than M may be evaluated by:

where is the threshold for non-linear collapse, which in the spherical model is given by

For an Einstein-de Sitter universe, . The mass distribution function is then easily derived:

Unfortunately, this result predicts that only half of the mass of the universe ends up in virialized objects, whereas one expects that at each epoch in a hierarchical scenario all of the mass should be bound in virialized objects. Press & Schechter (1974) side-stepped this problem by arbitrarily multiplying this mass function by a factor of 2. Several authors have recently re-examined the issue and proposed solutions (Peacock & Heavens 1990, Cole 1991, Blanchard et al. 1992a).

The problem has also been considered by Bond et al. (1991). Instead of smoothing the density contrast with a filter , they adopted a low-pass, top-hat filter in Fourier space. At a given point x in real space, the value of the smoothed density contrast executes a Gaussian random walk as the size of the filter is increased from to (corresponding to a decrease in the size of the window in real space). The step size of the random walk is a Gaussian random variable whose variance depends only on the power spectrum and on the value of (all of this applies only for Gaussian density fields). By identifying points which first pass the critical density at a particular value as objects of mass , the authors were able to recover the PS formula, including the troublesome factor of 2.

On the other hand, Blanchard et al. (1992a) point out that with the usually adopted simplifications (i.e. the collapse being driven essentially by the mean local density and the neglect of fragmentation), it is always possible to write the mass function in an exact way as:

where represents the fraction of volume covered by non-linear spheres of radius greater than R (the smoothing scale corresponding to mass M). Although the function F is well defined, its calculation represents an extremely complicated statistical problem for Gaussian random fields. It would appear, then, that the normalization problem of the PS formula is due to the assumption that only points which are at the center of non-linear spheres of radius R are counted, rather than all points (not just the centers) residing in spheres of radius R or greater.

We justify our use of the PS formalism by its surprisingly good fit to the mass functions found in numerical simulations. This was first emphasized by Efstathiou et al. (1988), who examined the cluster multiplicity function for different values of n. Let us consider the more recent simulations of White et al. (1993). With an error of less than 50% in the mass, the simulation distributions agree with the PS formula for masses above , where the abundance is . We may take this to mean that the PS mass function is reliable into the regime where it accounts for of the total mass. Roughly speaking, real clusters with a mass of a couple of have an abundance of a few and thus represent about of the total mass. For clusters with keV this means an abundance of . We conclude that the PS formula can be applied to confidently predict the abundance of X-ray clusters over the full range of observed temperatures.

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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