4.1 The SZ effect

In addition to the primary CMB anisotropies discussed so far, we also expect secondary anisotropies due to the interaction of CMB photons with clusters along the line of sight. The best known (and observed) is the Sunyaev–Zel’dovich (SZ) effect, which is due to the upscattering of CMB photons by electrons in the hot gas (7 8 Te ∼ 10 − 10 K) at the centre of clusters. Other secondary anisotropies include, for example, the Rees-Sciama effect, which is a result of a cluster collapsing during its encounter with a CMB photon.

In this review, we shall consider only the SZ effect, since this the only secondary anisotropy to have been observed to date. The SZ effect is in fact made up of two separate effects: One is due to the bulk velocity of the cluster, and the other due to the thermal velocities of the electrons in the cluster gas. These are called the kinematic and thermal SZ effects respectively. The kinematic effect measures the cluster peculiar velocity, whereas the thermal effect can be used, in conjunction with images and spectra of its X-ray emission, to study the cluster gas. In particular, for a dynamically relaxed cluster, we can use the thermal SZ effect and X-ray data to estimate the physical size of the cluster, and hence its distance. This, in turn, yields an estimate of the Hubble constant H0.

If the cluster has a peculiar velocity vp along the observer’s line of sight, then the temperature of a CMB photon is Doppler-shifted by an amount

vp∫ (ΔT )kin = T0 c neσT dl, (5 )
where T0 is the temperature of the CMB, ne is the electron number density, σT is the Thomson scattering cross-section and the integral is taken along the line of sight. In terms of the Rayleigh–Jeans brightness temperature this becomes
{ } vp ∫ --x2ex--- (ΔTRJ )kin = T0 c neσT dl (ex − 1 )2 , (6 )
or as an intensity
∫ { } 2k3T-30 vp --x4ex--- (ΔI ν)kin = h2c2 c ne σT dl (ex − 1)2 , (7 )
where we have set x = hν∕kT0.

The effect due to thermal motions of the electrons is second-order in the electron velocity, and does not preserve the blackbody shape of the CMB spectrum. For the thermal SZ effect the change in the Rayleigh–Jeans brightness temperature is given by

∫ kTe { x2ex ( x ) } (ΔTRJ )thermal = T0 neσT ----2 dl --x----2-[x coth -- − 4] , (8 ) mec (e − 1 ) 2
where Te is the electron temperature and me the electron mass. In terms of intensity this becomes
{ } k3T03∫ kTe x4ex ( x) (ΔI ν)thermal = 2--22- neσT ---2-dl --x-----2[xcoth -- − 4] . (9 ) h c mec (e − 1) 2
It is usual to describe the magnitude of the thermal effect in terms of the y parameter, which is given by
∫ -kTe- y = neσT m c2 dl, e

The frequency dependencies of the kinetic and thermal effects (i.e. those functions in curly brackets in Equations 6View Equation9View Equation), are shown in Figure 8View Image.

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Figure 8: The frequency dependences of the thermal and kinetic SZ effects expressed as a brightness temperature change (top) and intensity change (bottom).

Note that at ν = 210 GHz, the maximum change in intensity due to the kinematic effect coincides with the null of the thermal effect. This, in principle, allows one to separate the two effects. The magnitude of the thermal effect for a hot, dense cluster is (ΔTRJ )thermal ≈ 1 mK, and for reasonable cluster velocities the kinematic effect is an order of magnitude smaller.

Observations of the SZ effect have been made in Cambridge using the Ryle telescope, which is an 8-dish interferometer operating at 15 GHz [80], in Caltech using the Owen’s Valley 5.5 m telescope [61], the Owen’s Valley 40 m telescope [41] and the Owen’s Valley Millimeter Array (OVMMI) [14] [13], at NASA using the MSAM balloon experiment [84], at the Caltech Submillimeter Observatory using a purpose built instrument called the Sunyaev–Zel’dovich Infrared Experiment (SuZIE) [39] and by various other groups. The magnitude of the observed SZ effect in these clusters can be combined with X-ray data from ROSAT and ASCA to place limits on H0. This has been reviewed in Lasenby & Jones [52].

Another important feature of the SZ effect is that the decrement does not change with redshift. Therefore, it should be possible to detect clusters out to very high redshift. To test this the Ryle telescope has been making observations towards quasar pairs. Figure 9View Image

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Figure 9: Ryle telescope map of the sky towards quasar pair PC1643+4631 (the positions of which are shown as crosses). The central decrement is about − 600 μK.

shows the Ryle telescope map of the sky towards the quasar pair PC1643+4631 (Jones et al. 1997) [46]. These two quasars are at a redshift of ∼ 3.8 and are ∼ 3′ apart on the sky. They are a strong candidate for a gravitational lensed object due to the similarity in their spectra. As there is no X-ray detection with ROSAT the cluster responsible for the lensing must be at a redshift greater than 2.5 and from modelling of the gravitational lensing, or from fitting for a density profile in the SZ effect, it must have a total mass of about 15 2 × 10 M ⊙.

With observations of distant clusters it is possible to predict the mass density of the Universe. Using the Press–Schecter formalism the number of clusters in terms of SZ flux counts can be predicted. Figure 10View Image shows the results from Bartlett et al. [6]. From this figure it is seen that, if the decrements are really due to the SZ effect and there was no bias in selecting the fields (e.g. the field was chosen because of the magnification of the quasar pair images), then the Ω = 1 model of the Universe is ruled out. An open model is required to be consistent with the data. Confirmation of the detections are needed. Until these follow up observations have been made, it is impossible to say how accurate these findings are.

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Figure 10: SZ source counts with observational constraints, as a function of SZ flux density expressed at 400 GHz. The two hatched boxes show the 95% one-sided confidence limits from the VLA and the RT; due to the uncertain redshift of the clusters, there is a range of possible total SZ flux density, which has for a minimum the value observed in each beam and a maximum chosen here to correspond to z > 1. From the SuZIE blank fields, one can deduce the 95% upper limit shown as the triangle pointing downwards (Church et al. [18]). We also plot the predictions of our fiducial open model (Ω = 0.2) for all clusters (dashed line) and for those clusters with z > 4.

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