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Astron. Astrophys. 317, 1-13 (1997)

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2. The theoretical temperature distribution function

2.1. The mass function

The formation of nonlinear bound objects is expected to occur from the collapse of initial density fluctuations that grow under the influence of gravity. In the following, these fluctuations are assumed to obey gaussian statistics, although in the low- [FORMULA] case the motivation for non-gaussian fluctuations is higher. The case of non-gaussian fluctuations has been investigated in OBB and the difference with the gaussian case was found rather weak in front of changes introduced by changing [FORMULA].

Non-linear structures of mass M are expected to form from initial density fluctuations of comoving size [FORMULA] that contain mass M. We parameterize the initial mass power spectrum on galaxy clusters scale by a power law:

[EQUATION]

The variance of the initial density contrast field [FORMULA], smoothed with some window function at scale [FORMULA], is then:

[EQUATION]

In the case of a top-hat function, the window mass is

[EQUATION]

In linear theory, the initial rms density contrast extrapolated to the present epoch ( z=0 ) is:

[EQUATION]

In this equation, D is the growing mode of the linearized equation for the growth of perturbations and [FORMULA] is the redshift corresponding to an early time at which the fluctuations in the universe were still linear. In the case of [FORMULA] and for a vanishing cosmological constant [FORMULA], [FORMULA] with a being the expansion factor. At z=0 and for [FORMULA], Peebles (1980) gives a good analytical fit for [FORMULA]: [FORMULA]. Lahav et al. (1991) have extended the result to [FORMULA] and a non-zero cosmological constant case. However, in the case of our interest, a low density open universe, the expression can be analytically calculated (Weinberg 1972) and it is given in the Appendix. At very high redshifts, the behavior of the growth factor in the critical universe and in the open universe are similar. Due to the more rapid expansion of the scale factor, the growth of the perturbations at intermediate redshifts is much slower in the open universe than in the critical universe. This implies a fundamental difference between open and critical models: the formation of structures is expected to occur much earlier in open models.

The exact condition for the non-linear collapse of a structure in a hierarchical theory is a rather complicated question. One generally uses the spherical model which greatly simplifies the problem: it appears from N-body simulations (see for instance Thomas & Couchman 1992) as well as from analytical arguments (Bernardeau 1994) that it is a good description of the non-linear evolution of a perturbation. Within this model, we can calculate the critical overdensity [FORMULA] that an object collapsing at redshift z would have at redshift z=0 in the linear regime. The derivation of [FORMULA] for open universes is given in the Appendix.

Considerable attention has been devoted in the literature to the problem of the determination of the mass function. Most of these works were done in the case of the critical universe. However, due to the similarities in the set down problem, we can extrapolate the results to the low- [FORMULA] case. Using the spherical model, Press & Schechter (1974) proposed an estimate of the fraction of the mass of the universe that is in virialized objects. In the case of a gaussian random field, their mass distribution function is given by:

[EQUATION]

Although the PS derivation contains some uncertainty, Blanchard et al. (1992) argued that it is physically motivated and likely to give a good approximation of the mass function. In practice, the use of the PS formalism is justified by its amazingly good fit to the mass functions found in numerical simulations. This result was first emphasized by Efstathiou et al. (1988) who compared the multiplicity function of collapsed structures in the case of the critical universe and for different values of n, to the PS mass function. From their simulations we can consider that the PS formula is valid for objects which contribute to less than [FORMULA] of the total density. In the case of galaxy clusters whose masses are of the order of [FORMULA], this density corresponds to a space density close to [FORMULA]. Recently, the PS formalism has also been compared to N-body simulations in the case of a low density, zero curvature model (White et al. 1993). The predictions of the theory agree quite well with the results of the simulations down to an abundance of [FORMULA] ( h=0.5 ). As [FORMULA] only plays a role in the growing rate of the fluctuations and does not affect the dynamics at z=0 , we take the above simulations that included a cosmological constant, for an additionnal justification of the PS formalism down to the same limits.

2.2. The T-M and the [FORMULA] relations

In order to test the mass function on galaxy cluster scale, we need to relate the cluster mass to some observed quantity like the X-ray temperature or the X-ray luminosity. The X-ray emission of galaxy clusters is produced by thermal bremsstrahlung from the hot, optically thin intracluster plasma which is shock heated during the infall onto the collapsing structure (Evrard 1990a, 1990b). It is usually assumed that the intracluster gas is isothermal and that it is in hydrostatic equilibrium within the potential of the cluster. For more details concerning the distribution of the gas under this hypothesis, see OBB. In this paper, we restrict ourselves to changes introduced by the low [FORMULA] case.

From the equation of the hydrostatic equilibrium, one can relate the X-ray temperature to the virial mass,

[EQUATION]

In this equation, [FORMULA] and [FORMULA] are the mean molecular weight and the proton mass respectively and [FORMULA] is a parameter coming from the observed surface brightness profiles of clusters. The virial mass [FORMULA] and the virial radius [FORMULA] are quantities evaluated within a region where virialization has taken place. This regime is supposed to be achieved when the radius of the fluctuation is half of its maximum value. Assuming the spherical top-hat model, the density contrast [FORMULA] of such a region can be derived at any redshift (cf. to the Appendix). The virial radius is then:

[EQUATION]

Using the typical value of [FORMULA], [FORMULA] (Jones & Forman 1984), and taking into account the incomplete thermalization of the gas which appears from the hydrodynamic simulations performed by Evrard (1990a, 1990b), the relation between the temperature and the virial mass becomes:

[EQUATION]

where [FORMULA] is the cluster virial mass in units of [FORMULA]. The incomplete thermalization represents an approximately [FORMULA] lowering in the normalization of the above relation. However, the shape of this relation is quite well reproduced by the simulations (Evrard 1990a). It should be noticed that because of the frequency shift in an expanding universe, the observed or apparent temperature does not depend on redshift [FORMULA].

It would be extremely useful to derive a similar relation for the X-ray luminosity. However, the bolometric X-ray luminosity is given by:

[EQUATION]

which depends on the gas density profile, and in particular, it is very sensitive to the core density which produces most of the X-ray luminosity. Since the formation of the X-ray core radius is determined by complicated competing processes (cooling flow, galactic feedback) which are not yet well understood, the [FORMULA] relation is not easy to model. Indeed, a simple self-similar scaling (Kaiser, 1986) does not fit the observed local [FORMULA] relation (OBB and references therein). Therefore, throughout this paper we use:

[EQUATION]

which comes from the observations at z=0 (Edge & Stewart, 1991). Since this expression represents the bolometric X-ray luminosity, we must correct it for the fraction

[EQUATION]

which is collected in the interesting energy band [ [FORMULA] - [FORMULA] ] (in this equation we have assumed that the Gaunt factor is equal to 1).

Concerning the evolution of the [FORMULA] relation with redshift, Kaiser (1991) and Cavaliere et al. (1993) assert that clusters of a given temperature must have lower luminosities at higher redshifts in order to reproduce the negative evolution of the luminosity function (we shall return to this point beyond). However, Henry et al. (1994) claim that the evolution of cluster temperatures is moderate up to a redshift of 0.33 for clusters whose luminosity is close to [FORMULA]. Therefore, unless we specify it, we assume that relation 3does not depend on redshift. We will develop the implications of such an assumption within the last section.

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