2. The galaxy mass function
Mass condensations may be characterized by two parameters, such as for instance velocity dispersion and radius, or equivalently mass and radius. As will be explained in Sect. 3.1, we define galaxies as halos of mass M with a radius given by a condition provided by virialization and cooling constraints. This can be written as a mean density contrast within the radius , so that we have a one parameter family of halos. The relation will also depend on redshift, but since all halos we shall consider will have already virialized we have for all z and , and for any mass M: .
2.1. Press-Schechter approach (PS)
We shall first devise a "Press-Schechter approach" (PS), in the sense that we wish to recognize in the early universe, when overdensities still grow according to linear theory, the density fluctuations which will eventually become halos of mass M, density contrast , at the redshift of interest z. When this density contrast is constant and given by the virialization condition (obtained from the spherical collapse model) this is simply the usual Press-Schechter approximation.
Let us first consider the case . According to the spherical model, an overdensity characterized by the density contrast given by the linear theory, , will slowly separate from the general expansion, its radius will reach a maximum at time and collapse at time when . We shall assume that in fact the halo virializes at this collapse time at the radius , as implied by energy conservation and virial equilibrium if kinetic energy is negligible at the turn-around. Hence at time the density contrast of the halo is . Then we assume that the density of this object does not evolve significantly so that . Thus, a given halo of mass M, at redshift z, with a density contrast , collapsed at the redshift such that and the density contrast attached to this halo by the linear theory in the present universe is . If we note , where is the amplitude of the density fluctuations extrapolated to by linear theory as usual, we get:
as a function of mass and redshift. Then, as long as , we have
Then we consider that each halo is characterized by its parameter so that the mass fraction in objects between and is:
where we assumed gaussian initial fluctuations and we corrected by a factor 2 as in the traditional Press-Schechter prescription. Then the comoving multiplicity function of dark matter halos is:
where is the mean density of the present universe. When the density contrast is constant and equal to the formulation (4) is exactly the usual Press-Schechter multiplicity function. In the case , we have where is the growing mode of the linear theory and the density contrast of a halo which virializes at redshift is . Hence we now get:
where is defined by:
The multiplicity function of dark matter halos is still given by (4). Note that we have the normalization condition:
2.2. Non-linear hierarchical scaling model
We shall now devise a second method which deals directly with the non-linear regime. We define for each halo the parameter x by:
is the average of the two-body correlation function over a spherical cell of radius R. Then we write for the multiplicity function of these halos at a given redshift z (see VS):
while the mass fraction in halos of mass between M and is:
The function is a universal function that depends only on the initial spectrum of fluctuations and that has to be taken from numerical simulations although its qualitative behaviour is well-known: for small x with and for large x with and to 20. Moreover, it satisfies:
It has been shown in VS that this function is quite close to a similar scaling function that is obtained from the counts in cells. Bounds to estimate the difference between these two functions are given in VS while numerical checks are presented in Valageas et al. (1999a). Hereafter we use the function in place of . The correlation function that measures the non-linear fluctuations at scale R can be modelled in a way that follows very accurately the numerical simulations (see VS for more details).
2.3. Comparison of the PS model with the non-linear hierarchical scaling approach
These two formulations look very different, especially for evolution effects. The power at which redshift enters the expressions (4) and (9) is different even in the exponential. It happens, however, that the parameter of the linear theory can be expressed in terms of the parameter x of the non-linear scaling theory. We will shortly summarize this calculation for . A detailed account can be found in VS. Using an analytic expression for the evolution of the correlation function, one can write the non-linear correlation in the stable clustering regime as a function of the extrapolation of the linear correlation function to the epoch under consideration:
where is a parameter close to unity. Using this relation one can obtain for a power-law initial spectrum of index n:
which shows (VS) that the expression (4) deduced from linear theory using the Press-Schechter ideas can be brought into a form quite similar to the non-linear scaling one (9) with:
This holds in the highly non-linear regime, that is for and . The definition of galaxies we shall use, which considers that galaxies are objects which have already virialized (and have since undergone a significant cooling) ensures that we always have for all galaxies at any time. The condition implies that (14) will hold until the redshift such that . This similar scaling as a function of the parameter x, for , explains at the same time the similarity of the numerical estimates done with the two mass functions, but also their difference since the scaling functions and are definitely not the same . The coefficient in the exponential is to 0.1 in the non-linear scaling case whereas it is in the PS approach, which implies a considerably faster fall-off at large x. The powers of x, typically larger than unity, entering the exponent in (14) further increase this difference. Thus, as compared to the non-linear scaling mass function the PS prescription overpredicts the number of intermediate objects ( or ) and gives fewer extreme halos (very small or very massive objects), see VS for details. For the non-linear scaling approach will give more numerous bright galaxies than the PS prescription, but fewer ones at . A detailed comparison with numerical simulations of the various mass functions one can define within the framework of the non-linear scaling model, and with the PS prescription, is presented in Valageas et al. (1999a).
© European Southern Observatory (ESO) 1999
Online publication: April 19, 1999