At the same time, the electrons of the hot gas in the cluster Compton upscatter photons from the CMB radiation. At radio frequencies below the peak of the Planck distribution, this causes a “hole” in radio emission as photons are removed from this spectral region and turned into higherfrequency photons (see Figure 8). The decrement is given by an opticaldepth equation,
involving many of the same parameters and a function which depends on frequency and electron temperature. It follows that, if both and can be measured, we have two equations for the variables and the integrated length through the cluster and can calculate both quantities. Finally, if we assume that the projected size of the cluster on the sky is equal to , we can then derive an angular diameter distance if we know the angular size of the cluster. The Hubble constant is then easy to calculate, given the redshift of the cluster.


Although in principle a clean, singlestep method, in practice there are a number of possible difficulties. Firstly, the method involves two measurements, each with a list of possible errors. The Xray determination carries a calibration uncertainty and an uncertainty due to absorption by neutral hydrogen along the line of sight. The radio observation, as well as the calibration, is subject to possible errors due to subtraction of radio sources within the cluster which are unrelated to the SZ effect. Next, and probably most importantly, are the errors associated with the cluster modelling. In order to extract parameters such as electron temperature, we need to model the physics of the Xray cluster. This is not as difficult as it sounds, because Xray spectral information is usually available, and line ratio measurements give diagnostics of physical parameters. For this modelling the cluster is usually assumed to be in hydrostatic equilibrium, or a “betamodel” (a dependence of electron density with radius of the form n(r) = n_{0}) is assumed. Several recent works [137, 15] relax this assumption, instead constraining the profile of the cluster with available Xray information, and the dependence of H_{0} on these details is often reassuringly small ( 10% ). Finally, the cluster selection can be done carefully to avoid looking at cigarshaped clusters along the long axis (for which ) and therefore seeing more Xrays than one would predict. This can be done by avoiding clusters close to the flux limit of Xray fluxlimited samples, Reese et al. [121] estimate an overall random error budget of 20 – 30% for individual clusters. As in the case of gravitational lenses, the problem then becomes the relatively trivial one of making more measurements, provided there are no unforeseen systematics.
The cluster samples of the most recent SZ determinations (see Table 2) are not independent in that different authors often observe the same clusters. The most recent work, that in [15] is larger than the others and gives a higher H_{0}. It is worth noting, however, that if we draw subsamples from this work and compare the results with the other SZ work, the H_{0} values from the subsamples are consistent. For example, the H_{0} derived from the data in [15] and modelling of the five clusters also considered in [69] is actually lower than the value of 66 km s^{–1} Mpc^{–1} in [69].
It therefore seems as though SZ determinations of the Hubble constant are beginning to converge to a value of around 70 km s^{–1} Mpc^{–1}, although the errors are still large and values in the low to midsixties are still consistent with the data. Even more than in the case of gravitational lenses, measurements of H_{0} from individual clusters are occasionally discrepant by factors of nearly two in either direction, and it would probably teach us interesting astrophysics to investigate these cases further.
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