We now considered sources of error which may arise in our determination of through inaccuracies in our model.
We know from X-ray observations that, on large scales, the intra-cluster gas is typically very close to being isothermal out to radii of 750 kpc (Mushotsky (1996)). However, X-ray observations are insensitive to the low density gas far from the cluster centre (see equation 2), and so do not give us information on the temperature distribution of ``halo'' gas surrounding the central region.
If the temperature and density of the halo change in a manner such that the gas remains in hydrostatic equilibrium with the central region, then the line of sight pressure integral will be identical to that predicted by a purely isothermal King model. Thus the S--Z effect will be exactly that predicted by our simulations, and we will correctly estimate .
We have performed simulations to quantify the effect of hydrostatic equilibrium not being preserved in a gas halo. In one extreme example we assume that the gas density is unchanged from that predicted by the King model (equation 4) but allow the temperature to increase steadily to 3 times its central value beyond a radius of 750 kpc. Although this increases the temperature decrement which would be observed by a single dish telescope by , the S--Z flux that we would measure with the RT shortest baselines would change by only . The reason for this is that the RT is an interferometer and so resolves out any structure much larger than its synthesised beam, such as the changes in the line of sight pressure integral due to gas halos.
We therefore conclude that the exact nature of the outer regions of the intracluster gas does not significantly affect our determinations.
In many clusters the gas density is sufficiently high at the centre that the radiative cooling time is less than the age of the cluster (Fabian et al. (1991)). To maintain pressure balance, this cool gas collapses inwards and becomes denser. These cooling flows have now been detected in most rich clusters through the greatly increased X-ray emission of the cool, dense, central gas.
The cooling flow radius is typically of the order of 100 kpc; outside this region the gas remains isothermal. We therefore blank the pixels at the centre of the X-ray map and fit an isothermal model to only the regions where there has not been any significant cooling. If quasi-hydrostatic equilibrium is maintained in the central regions where cooling flows are formed, then through similar arguments to those used in section 4.1, the magnitude of the S-Z effect in such a cluster is identical to that predicted by a purely isothermal model. Thus we will be able to accurately estimate in clusters with quasi-hydrostatic cooling flows. Further, we have performed simulations with worst-case, non-hydrostatic cooling flow models which indicate that even in these clusters the effect on the expected flux measured by the RT would be negligible.
It is possible that although clusters appear to be isothermal on large scales, there may be temperature structure on small scales, below the resolution of X-ray telescopes; the intracluster medium may consist of a mixture of cold, dense and hot, diffuse clumps, still in hydrostatic equilibrium, and with thermal conduction suppressed by magnetic fields. Modelling such a cluster assuming that it was isothermal would incorrectly estimate both the gas temperature and the X-ray surface brightness. However, these two errors tend to cancel each other when calculating . Using a simple two-phase model of a clumped cluster where a volume fraction has density and the remaining has density 1, we find that the fractional error in resulting from modelling the cluster as being isothermal is given by
Thus for a cluster where half the gas volume has double the temperature of the other half (), the error in estimating is . An extreme case where a tenth of the gas mass occupies only a hundredth of the volume () leads to a error; higher levels of clumping than given in this model could easily be detected from the cluster's X-ray spectrum (Edge, private communication).
In calculating we have assumed that the line of sight depth, , through the cluster is equal to the width in the plane of the sky, . If this is not the case then the calculated value of Hubble's constant, will be related to its real value by
X-ray maps show that clusters are elliptical in the plane of the sky, with ellipticities of being common. Therefore to obtain a robust estimate of we must observe an orientation-unbiased sample of clusters. This sample can be compiled from a X-ray catalogue by selecting clusters above a certain luminosity limit (rather than a surface brightness limit). At present we are working on such a sample derived from the ROSAT All Sky Survey. The true value of is then the geometric mean of the individual estimates.
Combining X-ray and S--Z data depends on the cosmological deceleration parameter as well as (equation 3 assumes that ). In theory, observing two clusters will yield both of these parameters. We have estimated the possible error in calculating from assuming that by simulating the response of the RT to a rich cluster (; ; ; ) at redshifts between 0.1 and 10 for and 0.5. The results are plotted in Figure 4.
Figure 4: Predicted S--Z flux density on RT shortest baseline when observing a rich cluster with for different values of the cosmological deceleration parameter, .
It can be seen that the value of adopted makes little difference to the predicted flux observed by the RT, especially between redshifts of 0.1 and 0.5. It is also interesting to note that Figure 4 implies that if rich clusters exist in the early universe, then we should be able to detect them with the RT out to redshifts of 10. will, however, affect our fit to the X-ray emission from the cluster. We calculate that at the change in our estimate of between assuming and is only . The error rises to for a cluster at and to at .