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4.2 The Sunyaev–Zel’dovich effect

The basic principle of the Sunyaev–Zel’dovich (S-Z) method [147], including its use to determine the Hubble constant [141], is reviewed in detail in [12Jump To The Next Citation Point20]. It is based on the physics of hot (108 K) gas in clusters, which emits X-rays by bremsstrahlung emission with a surface brightness given by the equation
1 ∫ bX = ---------3- n2eΛe dl (14 ) 4π(1 + z)
(see e.g. [12]), where ne is the electron density and Λe the spectral emissivity, which depends on the electron temperature.

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 8View Image). The decrement is given by an optical-depth equation,

∫ ΔI (ν ) = I0 neσT Ψ(ν,Te )dl, (15 )
involving many of the same parameters and a function Ψ which depends on frequency and electron temperature. It follows that, if both bX and ΔI (x ) can be measured, we have two equations for the variables n e and the integrated length l ∥ through the cluster and can calculate both quantities. Finally, if we assume that the projected size l⊥ of the cluster on the sky is equal to l∥, 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.
View Image

Figure 8: S-Z decrement observation of Abell 697 with the Ryle telescope in contours superimposed on the ROSAT grey-scale image. Reproduced from [69Jump To The Next Citation Point].

Reference Number of clusters Model type
H0 determination
[km s–1 Mpc–1]

[15Jump To The Next Citation Point] 38 β + H 76.9+−33..94+−180..00
[69Jump To The Next Citation Point] 5 β +11+9 66− 10− 8
[165] 7 β 67+−301+8−156
[137Jump To The Next Citation Point] 3 H 69 ± 8
[94] 7 β 66+−141+1−1515
[121Jump To The Next Citation Point] 18 β 60+4+13 − 4−18

Table 2: Some recent measurements of H0 using the S-Z effect. Model types are β for the assumption of a β-model and H for a hydrostatic equilibrium model.

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) = n02 2 −3β∕2 (1 + r ∕rc)) is assumed. Several recent works [13715Jump To The Next Citation Point] 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 l⊥ ⁄= l∥) 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 [15Jump To The Next Citation Point] 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 [69Jump To The Next Citation Point] 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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