The "standard" picture for the origin of the large-scale distribution of matter in the universe is based on the gravitational growth of initially small density perturbations assumed to be present from the very earliest moments of cosmic time (Lemaître 1933). This idea has received considerable attention from theorists and recently some spectacular observational support: the detection of temperature fluctuations in the Cosmic Background Radiation (CBR) by the COBE instrument DMR (Smoot et al. 1992). To this discovery one may today add a host of other claimed detections (see, for example, Scott et al. 1995). In the gravitational instability scenario, the density field can be quantitatively described by the power spectrum of the perturbations and by their higher order moments. In this paper, although the non-gaussian case will be briefly discussed, we will be primarily concerned with Gaussian perturbations for which the power spectrum provides a sufficient description. The uncertain origin of the density perturbations translates into an inability to calculate from first principles the form of the power spectrum, and the lack of a specific value for the density parameter exacerbates the problem. The theory of inflation alleviates this theoretical uncertainty by predicting that the primordial spectrum follows a power-law: with , the exact value of depending upon the possible existence of gravitational waves (Lucchin et al. 1992). In addition, inflation sets the cosmic density to the critical value (, where is the current value of the Hubble constant). Once the nature of the (necessarily non-baryonic) dark matter is specified, one may calculate the final, evolved power spectrum as a function of the amplitude and index of the initial spectrum. This is the spectrum relevant to galaxy formation.
Among the possible scenarios based on inflation, the cold dark matter (CDM) model has proven very successful in explaining many observed properties of the universe on scales ranging from galaxies to galaxy clusters, i.e. between a few tens of kpc and a few Mpc (for a review on the subject, see Frenk 1991). The amplitude of the power spectrum is the only free parameter of the "standard" model: one adopts .
The galaxy cluster population provides some of the most stringent constraints on models of galaxy formation, essentially because clusters are rare objects and, hence, their properties are sensitive to the underlying density fluctuations. The obviously important role of baryons in the determination of the observed properties of clusters would seem to necessitate hydrodynamical simulations in order to derive any such constraints. Evrard (1989) has pioneered this interesting and important approach. Nevertheless, the ensemble properties of clusters, like their optical and X-ray luminosity functions, or their velocity and temperature distribution functions, are difficult to address directly by numerical simulations because the size of the numerical box must be very large in order to contain a sufficient number of clusters; an analytical approach remains an effective alternative.
Kaiser (1986) made an important theoretical step in this direction by proposing and then using simple scaling laws for cluster properties to derive the evolution of the ensemble properties. The corroboration of the Press & Schechter (1974) (PS) mass function by some recent numerical simulations (Efstathiou et al. 1988, Carlberg & Couchman 1989, Gelb & Bertschinger 1994, Eke et al. 1996; see Brainerd & Villumsen 1992 for an alternate point of view) provides us with an even more powerful tool for constraining galaxy formation theories: the simple PS formula gives us the mass function of structures at any redshift for any theory of initially Gaussian fluctuations. Provided that the relation between some observed quantity (for instance, the luminosity) and the mass is known, perhaps provided by hydrodynamical simulations of a small number of clusters, both the cluster ensemble properties (e.g. the luminosity function) and their evolution can be predicted and compared to observations. For example, Schaeffer and Silk (1988) showed that the CDM scenario reproduces well the optical luminosity function of galaxy clusters. Evrard (1989), using the number density of galaxy clusters with velocity dispersions greater than at redshifts lower than 0.1, concluded that the bias parameter b - which is defined to be the inverse of the rms value of mass fluctuations within spheres of radius - must be of the order of 1.5, inconsistent with the higher values of b required to explain galactic properties. He pointed out that an even smaller value of the bias is necessary to explain the three high velocity dispersion clusters listed by Gunn (1989) at redshifts greater than 0.1. Peebles et al. (1989) reached a similar conclusion by using a variety of present day cluster properties. However, Frenk et al. (1990) argued, by constructing artificial cluster catalogues from numerical simulations, that the cluster velocity dispersion is not a property from which reliable constraints can be derived on models because projection effects along the line of sight can contaminate the galaxy samples and significantly increase the estimated velocity dispersion. Since then, the standard CDM model has met with some further serious problems, for example an inability to explain the angular correlations of galaxies detected with the Automatic Plate Measuring Machine (APM) (Maddox et al. 1990). This has shed doubt on the validity of "standard" CDM. In addition, the amplitude indicated by the COBE temperature fluctuations, corresponding to rather than the advocated , is generally considered too high for the model to be viable (see, however, Bartlett & Blanchard 1994, 1996), although some other authors have suggested that this high normalization can explain the observations once non-linear effects have been properly accounted for (Couchman & Carlberg 1992).
To solve the various problems faced by the standard version of CDM, changes to the power spectrum have been proposed, such as suppressing the small-scale power by mixing in a small amount of hot dark matter (Bond et al. 1980, Bond & Szalay 1983, Dekel 1984, Schaefer & Shafi 1992, Davis et al. 1992) or by altering the primordial value of (Cen et al. 1992, Cen & Ostriker 1993) (so-called "tilted" CDM models). A list of further other possibilities is given by McNally & Peacock (1995). All of this leads us to reconsider the form of the power spectrum and adopt the point of view that it is an unknown which we wish to constrain. This analysis constitutes an alternative to a direct analysis of galaxy surveys (Peacock & Dodds 1994) for which the amplitude of the bias is unknown and does not permit direct access to the mass distribution. In particular, we will use the cluster population for this purpose: we will assume that over cluster scales even the evolved power spectrum can be approximated by a power-law, and then we will use the ensemble cluster properties to place limits on the amplitude and spectral index n. The adoption of a power-law is not really restrictive as most currently considered models lend themselves to this approximation (this may not necessarily be the case in purely baryonic models, in which the Jeans mass may strongly influence the perturbations on cluster scales and in a manner dependent upon the ionization history).
In principle, both optical and X-ray data can be used to constrain models, but, as emphasized by Frenk et al. (1990), the optical properties are subject to projection effects. If indeed important, such effects can alter both the optical luminosity and the velocity distribution functions. Moreover, the relationship between the overall mass of a cluster and its constituent galaxies could very well be complicated by the non-linear physics of galaxy formation (Evrard et al. 1994). The X-ray properties of a cluster offer an interesting alternative as they should not suffer the same severe projection effects. However, the observed X-ray luminosity of clusters is dominated by their core radius, and the physical origin of this core is unknown. This makes it difficult to relate the X-ray luminosity to the cluster mass, a point we will discuss in greater detail below and which will lead us to focus on the temperature distribution function.
In recent years many authors have calculated the ensemble properties of X-ray clusters expected in various scenarios (Henry & Arnaud 1991, Blanchard & Silk 1991, Kaiser 1991, Pierre 1991, Lilje 1992, Oukbir & Blanchard 1992, Bahcall & Cen 1993, Bartlett & Silk 1993, Blanchard et al. 1994, Colafrancesco & Vittorio 1994, Balland & Blanchard 1995 Liddle et al. 1995, Eke et al. 1996, Pen 1996) and compared the results with observations (Edge et al. 1990, Henry & Arnaud 1991) in order to derive constraints on the power spectrum. Henry & Arnaud (1991) found that the spectral index n and the bias b of the density perturbations must be -2.1 and 1.7, respectively, to reproduce their observed temperature distribution function. Blanchard & Silk (1991) claimed that the CDM model is marginally consistent with the Edge et al. (1990) data if the bias parameter is close to 1.5, but that with is favored over the CDM value of on cluster scales. However, Kaiser (1991) argues that the observed evolution of the luminosity function needs an index closer to -1. Lilje (1992) has shown that , flat CDM models need to be antibiased in order to reproduce the temperature distribution function. He also noticed that at high redshifts the temperature distribution evolves differently depending on the value of . Both of these conclusions are consistent with the results of Bartlett & Silk (1993). Oukbir & Blanchard (1992, 1996) have shown that an unbiased open universe with is compatible with the observed temperature distribution and that the redshift distribution of X-ray clusters is a powerful test of the mean density of the universe. Colafrancesco & Vittorio (1994) investigated the constraints imposed by the cluster luminosity function on a variety of models normalized to COBE, extending the analysis of Bartlett & Silk (1993), although reaching quite different conclusions.
Given this large list of different analyses, it would seem difficult to derive a consistent set of constraints from observations of the cluster population. In this paper we re-examine the modeling of X-ray clusters to clarify the situation. We construct a self-consistent set of relations between observable cluster properties and the theoretically relevant virial mass. We then use these relations to obtain robust constraints on the power spectrum (i.e. on the amplitude and spectral index). These constraints are applicable on scales from 5 to 15 Mpc. In this work we consider only a flat, hierarchical, dark matter dominated universe with . We start in the next section with a presentation of the arguments supporting the PS formula. In the following section, we discuss the observed properties of individual clusters and then relate them to the mass appearing in the theoretical mass function (Sect. 3). In the fourth section we derive the power spectrum parameters which best reproduce the observed temperature distribution function. In the fifth section we dicuss the expected evolution of the luminosity function and compare the model to the high redshift observations. The sixth section presents predictions for the cluster number counts as a function of X-ray flux and an estimate of the cluster contribution to the soft X-ray background. Finally, the seventh section contains a brief discussion of non-Gaussian fluctuations. In the last section we summarize our results.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998