Astron. Astrophys. 336, 425-432 (1998)

1. Introduction

Most favored theories of structure formation in the Universe are based on the gravitational growth of initially small density perturbations with Gaussian statistics. The Gaussian characteristic finds its way into the mass function of cosmic structures and leads to the expectation of an exponentially rapid decline in the number density of objects with increasing mass (Press & Schechter 1974). The interesting consequence is that the abundance of massive objects, such as galaxy clusters, is then extremely sensitive to the power spectrum (amplitude and shape) of the density fluctuations and their growth rate with redshift. Since the growth rate is controlled by the density parameter of the Universe, , this means that the shape of the redshift distribution of clusters of a given mass is sensitive to . In fact, once constrained by local data, the redshift distribution depends only on the underlying cosmology, i.e., (and to a lesser extent, on the cosmological constant ). In other words, the redshift distribution provides a probe of the density of the Universe (Oukbir & Blanchard 1992, 1997; Blanchard & Bartlett 1998). The sense is such that we expect many more clusters at large redshift if , because the growth rate is suppressed by the rapid expansion at late times in open models, leading to less evolution towards the past.

Essential for the application of this probe of is the existence of an easily observed cluster quantity well correlated with virial mass. It has been cautioned by many authors that the velocity dispersion of cluster member galaxies is too easily inflated by contamination of interlopers along the line-of-sight (e.g., Lucy 1983; Frenk et al. 1990; Bower et al. 1997). Many have turned instead to the X-ray temperature of the intracluster medium (ICM). On theoretical grounds, it is believed that the gas is heated by infall to the virial temperature of the cluster gravitational potential well. Numerical simulations in fact support the existence of a tight relation between virial mass and X-ray, which is to say, emission weighted, temperature (Evrard 1990; Evrard et al. 1996). The dependence of the relation is as expected based on the idea that the gas is shock heated to the virial temperature on infall, although the simulations indicate that there is an incomplete thermalization of the gas, resulting in a temperature slightly ( %) smaller than the virial value. The X-ray temperature is to be preferred over the X-ray luminosity as an indicator of virial mass, because the X-ray luminosity depends not only on the temperature, but also on the quantity of gas and on its density, or, what is equivalent, the spatial distribution of the ICM. This spatial distribution is difficult to model, particularly because there is at present no understanding of the origin of the ICM core radius.

Use of the X-ray temperature function to constrain models of structure formation is rather well developed as a subject (e.g., see Bartlett 1997 and references therein). Temperature data on clusters at is just now becoming available, and the possibility of even higher z data from future space missions like XMM makes the application of the redshift distribution test proposed by Oukbir & Blanchard a real possibility over the near term (see, e.g., Sadat et al. 1998).

The Sunyaev-Zel'dovich (SZ) (Sunyaev & Zel'dovich 1972) effect offers another, complimentary approach to the problem of applying mass function evolution as a probe of . Due to the distance independence of the surface brightness of the distortion, the effect represents an efficient method of finding high redshift clusters. This should be contrasted with X-ray emission, whose surface brightness suffers the cosmological dimming. Moreover, as will be developed below, the SZ effect has other, important advantages over X-ray studies: the integrated SZ signal of a cluster, its flux density (measured in Jy), is proportional to the total hot gas mass times the particle weighted temperature. This means that the signal is independent of the gas' spatial distribution and that the temperature involved is closely tied to the cluster virial mass, by energy conservation during collapse, for it is simply the total energy of the system divided by the number of gas particles. For the same reason, this temperature should also be much less sensitive to any temperature structure in the gas than the X-ray (emission weighted) temperature. Thus, the SZ effect is an observable which combines ease of theoretical modeling with ease of detection at large z.

All of this has prompted several calculations of the expected SZ number counts and redshift distribution of SZ selected clusters, and their dependence on the cosmological parameters and ICM evolution (Korolyov et al. 1986; Bartlett & Silk 1994; Markevitch et al. 1994; Barbosa et al. 1996; Eke et al. 1996; Colafrancesco et al. 1997). The future of this kind of study appears bright with the prospect of the Planck Surveyor satellite mission (http://astro.estec.esa.nl/Planck/ ). Ground based efforts have also made astounding progress recently, and it is already feasible to map square degree of sky to produce number counts down to flux levels sufficient to test theories (Holzapfel, private communication).

In this paper, we discuss what may already be an indication of clusters at very large redshift and the resulting implications. We refer to two radio decrements, one found in a deep VLA field (Richards et al. 1997) and the other detected by the Ryle Telescope during an observation of a double quasar system (Jones et al. 1997). Although these detections await definitive confirmation, we will nevertheless proceed to outline here the implications of their explanation as the thermal SZ effect produced by two clusters. What makes just two such objects of great importance is the fact that no optical or X-ray counterparts have been observed, and the flux limits in the X-ray are so stringent that the clusters would have to be at large redshift (Richards et al. 1997; Kneissl 1997; Kneissl et al. 1998). This is of paramount importance because, as we have mentioned, massive clusters (say ) at large z are not expected in critical models. Our goal in this paper is to quantify just how badly critical models fare in this regard. We emphasize that the modeling is based on the observed characteristics of the galaxy cluster population, in particular the X-ray luminosity-temperature relation and constraints on its potential evolution. This excludes from the present discussion the possibility of a large class of low luminosity clusters (both optical and X-ray). We believe that a clear discussion in this restricted context is nevertheless useful. (For this reason we dubb this work a "tale"!). The procedure also demonstrates the great potential of SZ cluster searches for constraining theories of structure formation.

The plot of the tale proceeds as follows: In the next section, we introduce the two radio decrements and their properties which will be used later. Then we outline our modeling of the SZ cluster population and of the two radio decrements. This is followed by a discussion of the X-ray emission to be expected from clusters producing the observed SZ signals and the minimum redshifts imposed by the X-ray flux upper limits; this represents a key element of our tale. Finally, we discuss the results and various caveats in the analysis before bringing an end to the tale with a brief summary.

© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998
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