Figure 1: Redshift distribution for clusters of given SZ flux density in the two cosmological models -- critical (solid lines) and open with (dashed lines); see Table 1 for the model parameters. The curves are drawn for and , corresponding to the VLA and RT objects, respectively, when translated to . For clarity, the two curves for the critical model are not labelled explicitly. We adopt .
The Sunyaev--Zel'dovich effect offers unique advantages for finding high redshift clusters and quantifying their abundance. The surface brightness of a cluster relative to the unperturbed CMB is expressed as a product of a spectral function, , and the Compton -parameter, which is an integral of the electron pressure along the line-of-sight: . Integrating the surface brightness over solid angle yields the following functional form for the total flux density of a cluster with angular size :
where is the total, virial mass of the cluster, is the hot gas mass fraction of clusters and is the angular-size distance. In the last line, we have used the fact that there exists a tight relation between X-ray temperature and virial mass: (Evrard et al. (1996)). Let's compare this with the corresponding expression for the X-ray flux of a cluster:
with denoting the luminosity distance. By comparing these two expressions we see that, in contrast to the SZ flux density, the X-ray flux suffers cosmological surface brightness dimming, represented by the extra factors of in the denominator of Eq. (4) which convert the angular-size distance to the luminosity distance. Besides this well-known difference, which tells us that the SZ effect is the more efficient way to find high-redshift clusters, we note that the X-ray emission depends on the gas density in addition to the hot gas mass fraction and temperature. This is unfortunate, because it means that the X-ray flux from a cluster depends on the core radius and profile of the intracluster medium (ICM) -- two quantities which are poorly, if at all, understood from the theoretical point of view. The SZ flux density presents the important advantage that it depends only on the total gas mass and the temperature, and not on the ICM's distribution. It is also true that the temperature which appears in the expression for the SZ flux density is a simpler quantity than the X-ray measured temperature: it is the mean, particle-weighted energy of the gas particles instead of, as in the case of X-rays, the emission-weighted gas temperature. This SZ temperature is a quantity which should be all the more closely related to the virial mass of a cluster than even the X-ray temperature, and less affected by any temperature structure in the cluster.
Figure 2: SZ source counts with observational constraints, as a function of SZ flux density expressed at . The two hatched boxes show the 95% one-sided confidence limits from the VLA and the RT; due to the uncertain redshift of the clusters, there is a range of possible total SZ flux density, which has for a minimum the value observed in each beam and a maximum chosen here to correspond to . From the SuZIE blank fields, one can deduce the 95% upper limit shown as the triangle pointing downwards (Church et al. (1997)). We also plot the predictions of our fiducial open model () for all clusters (dashed line) and for those clusters with . The critical model has great difficulty explaining the observed objects even with a lower redshift cutoff of only ; the actual limit from the X-ray data is stronger, but this would fall well off to the lower left of the plot. We assume .
Now the game is clear: with Eq. (3) we may convert the mass function into a distribution of clusters in SZ flux density and redshift (the quantitative relation for can be found in Barbosa et al. (1996)). The redshift distribution of clusters and the total source counts are then easily calculable (Korolyov et al. (1986), Markevitch et al. (1994), Bartlett & Silk (1994), Barbosa et al. (1996), Eke et al. (1996), Colafrancesco et al. (1997)). In Fig.1, we show the redshift distribution for clusters of two given SZ flux densities and for two representative cosmologies -- a critical model and a model with and . For this calculation, we have used a constant gas mass fraction (Evrard (1997)). The two chosen fluxes are our estimates of the flux density of the VLA and RT objects, when translated to a wavelength of by using the SZ spectral function, ; this is our fiducial working frequency and corresponds to the peak of the SZ distortion.
The key aspect of this figure is that, at a given flux density, there is an enormous difference between the number of high-redshift clusters in a critical and open universe. It is for this reason that even the detection of only two SZ decrements warrants the present discussion, because they appear to be at large redshift. Let us now quantify this by comparing the predicted number counts of clusters with redshifts greater than some minimum value with the counts implied by the detection of these two objects. This is done in Fig.2. The observed counts have been estimated using Poisson statistics and the amount of sky coverage in each case -- for the VLA (two fields) and for the RT (three fields). These constraints are given as two boxes because there is actually a range of possible total SZ flux density from each object, due to the unknown redshifts: at low redshift, the objects would be resolved and their flux density has to be corrected upwards. The minimum flux density is clearly the value observed, while for the maximum, we give the values for assuming an isothermal . The limits on the counts (i.e., in the vertical direction) are generous in that they represent the 95% one-sided Poisson confidence limits. We also show an upper limit on the counts obtained by the SuZIE instrument (Church et al. (1997)), which found no objects in a survey area of down to the limiting flux shown; the symbol represents the resulting 95% confidence upper limit. Predictions for the number of clusters on the sky for the two cosmological models and with varying minimum redshifts are shown by the labelled curves.