For this paper: postscript 3 Calculation of ... Abstract... 5 Conclusions...

4 The Hubble diagram  

Hubble constant values have been given in the literature for seven clusters of galaxies with redshifts from 0.023 to 0.182: Abell 1656 (Herbig et al. (1995)); Abell 2256, 478, and 2142 (Myers et al. (1997)); Abell 2163 (Holzapfel et al. (1997)); Abell 2218 (McHardy et al. (1990), Birkinshaw & Hughes (1994), Jones (1995)); and Abell 665 (Birkinshaw et al. (1991)). With the addition of CL 0016+16 at redshift 0.546, the resulting Hubble diagram (Figure 2) covers a wide range of redshifts. Although the accuracy of the individual measurements is low, we can use the diagram to reach a best-guess value for of , with an error of about if the errors on the individual measurements are independent. No useful constraints can be placed on based on these data.

 
Figure 2:  The Hubble diagram based on distances derived using the Sunyaev-Zel'dovich effect and X-ray data. Lines show the theoretical curves for and . Three independent determinations of the distance are shown for Abell 2218 (which lies at redshift 0.171).

However this error on is unreliable, since the points on the curve are not fully independent. The random elements of the errors in the distance scale are (presumably) independent from cluster to cluster, but the distances rely on good knowledge of the three-dimensional shapes of the clusters and on well-calibrated flux and brightness temperature scales for the X-ray and SZ effects respectively.

The three-dimensional structure leads to an orientation-dependent error in the distance scale: it is easier to measure the X-ray emission and SZ effects from clusters with long axes lying down the line of sight (Birkinshaw et al. (1991)). This can introduce an error if the selection criteria for the clusters in the Hubble diagram are not orientation-independent: for some of the clusters used here this is not the case -- CL 0016+16, for example, was selected on the basis of optical imaging which, like the X-ray surveys, is better at finding clusters which are elongated on the line of sight or superpositions of clusters. Other model-dependent errors in the estimate for are likely to be less important, and are discussed in Birkinshaw (1997).

The X-ray flux scale of ROSAT is critical in these measurements, and this is presumably uncertain by 5 per cent or so, which will introduce a systematic 5 per cent error in all the distances. For the SZ effects, errors at the 5 per cent level in the brightness temperature scale are also possible: this will introduce a systematic 10 per cent error into all distance measurements made by a particular instrument. There is therefore some freedom at the level of 10 per cent or so to move the measurements of sets of clusters made by particular groups up or down on Figure 2, and the freedom to move the entire set which use ROSAT data up or down by about 5 per cent. It is thought that calibration errors in the X-ray spectroscopy are less important.