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

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1. Introduction

The determination of the density parameter, [FORMULA], has received much attention in cosmology. Theory favors a value of [FORMULA] equal to one, because inflation naturally leads to this value. However, observations indicate a much lower value of [FORMULA]. The contribution of the visible part of galaxies is [FORMULA] (Peebles, 1993). If massive galactic halos are taken into account, then the derived value of [FORMULA] is about 3 to 10 times higher. On larger scales, from dynamical estimates within galaxy clusters as well as from the cosmic virial theorem, the measured value of [FORMULA] is around 0.2 (for a review on estimates of the content of the universe, see White 1990). However, it must be emphasized that the latter dynamical derivations of the mean density are based on the assumption that the galaxy systems considered are representative of the universe. This is probably not the case: different scales imply different values of [FORMULA], with a tendency for the inferred mean density to increase with scale. On scales of a few tens of Mpc, the value derived from the peculiar velocity field is consistent with the critical value (Dekel 1993). The reason for this discrepancy is not yet fully understood. It is therefore of fundamental interest to estimate the density of the universe by methods which differ from the usual dynamical ones. The goal of our paper is to present such a possibility.

There are recent attempts to constrain the density parameter from the morphology of galaxy clusters. Assuming that the distribution of density perturbations at high redshifts is gaussian, Richstone et al. (1992) placed a lower limit on [FORMULA] from an estimate of the fraction of rich clusters with significant substructures. Their study is based on the expectation that most structures have settled into some stable state in a low density universe. However, such an argument relies on subtle dynamics in the non-linear regime and should be interpreted with caution. Although the conclusions are still preliminary, a similar argument based on the structure of the X-ray gas have also led to a high value of the density parameter of the universe (Mohr et al. 1995).

As we have mentioned, although most of the observations do not provide support for such a conclusion, many theorists have favored a value of [FORMULA] equal to 1, primarily because of the attractive features of inflationary models. Furthermore, inflation also predicts the shape of the power spectrum of the primordial density fluctuations generated during the very early universe and provided that the nature of the dark matter is specified, the amplitude of the power spectrum remains the only free parameter in these models (at least, in their simpler version). In the case of a low density open model, and in the absence of a standard scenario, the fluctuations are not well specified and the more conservative approach is to assume on a given scale range, a power spectrum of the form [FORMULA], where n is a free parameter. The index n as well as the normalization of the power spectrum are then to be constrained. The abundance of galaxy clusters provides a interesting tool to derive these parameters on scales of a few tens of Mpc, since clusters result from the collapse of fluctuations with typical comoving radius in this range.

For practical purposes, the Press & Schechter formalism (1974) (PS hereafter) gives an accurate determination of the mass function. Provided that the mass is related to some observed quantity like the X-ray temperature or the X-ray luminosity, a comparison with observations allows one to constrain the amplitude and the shape of the power spectrum on galaxy cluster scales. In the recent years, many authors have investigated for the case of a critical universe, the consequences of the observed X-ray temperature and X-ray luminosity distribution functions. Oukbir et al. (1996) (OBB hereafter) provide a comprehensive discussion of the subject and evaluate in detail the various constraints that can be inferred from such a study. The case of a low density universe has also been investigated, but restricted to the context of a flat universe or with the shape of the power spectrum given by the CDM theory (Efstathiou et al. 1992, Lilje 1992, White et al. 1993, Bartlett & Silk 1993, Liddle et al. 1995), or to address some specific topic (Oukbir & Blanchard 1992, Hattori & Matsuzawa 1995, Eke et al. 1996).

In this paper we extend these studies to the case of a low density open universe. In Sect. 2, we discuss the arguments supporting the PS formalism and the relations between the cluster mass, the X-ray temperature and the X- ray luminosity in low density universes. In Sect. 3, we constrain the shape and the amplitude of the initial mass power spectrum by using the observed temperature distribution function. Since the normalization of the power spectrum depends on [FORMULA], we also constrain the density parameter. The critical model and the open model both reproduce the observations at z=0 and cannot therefore be distinguished by mean of present day data. However, in open low density universes structure formation is occuring much earlier than in flat models. Therefore, we also examine the observational constraints implied by high redshift clusters. In Sect. 4, we investigate the redshift distribution of X-ray temperature selected galaxy clusters. We show that this distribution provides a robust information on the mean density of the universe independently of the power spectrum of the fluctuations. During the preparation of this work, the redshift distribution of the Einstein clusters has been established. We have then examined the possibility of constraining the density parameter from these data, despite the fact that we do not have information on the temperature of these clusters.

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