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
higher-frequency photons (see Figure 8). The decrement is given by an optical-depth equation,
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Although in principle a clean, single-step method, in practice there are a number of possible difficulties.
Firstly, the method involves two measurements, each with a list of possible errors. The X-ray 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 S-Z 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 X-ray cluster. This is not as difficult as it sounds, because X-ray 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 “beta-model” (a
dependence of electron density with radius of the form n(r) = n0) is assumed.
Several recent works [137, 15
] relax this assumption, instead constraining the profile of the
cluster with available X-ray information, and the dependence of H0 on these details is often
reassuringly small (
10% ). Finally, the cluster selection can be done carefully to avoid
looking at cigar-shaped clusters along the long axis (for which
) and therefore seeing
more X-rays than one would predict. This can be done by avoiding clusters close to the flux
limit of X-ray flux-limited 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 S-Z 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 H0. It is worth noting, however, that if we draw
subsamples from this work and compare the results with the other S-Z work, the H0 values from the
subsamples are consistent. For example, the H0 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 S-Z 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 mid-sixties are still consistent with the data. Even more than in the case of gravitational lenses, measurements of H0 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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