4. The thermal Sunyaev-Zel'dovich effect

The results in Section 3 indicate that passage of radiation through an electron population with significant energy content will produce a distortion of the radiation's spectrum. In the present section the question of the effect of thermal electrons on the CMBR is addressed in terms of the three likely sites for such a distortion to occur:

1.	the atmospheres of clusters of galaxies
2.	the ionized content of the Universe as a whole, and
3.	ionized gas close to us.

4.1. The Sunyaev-Zel'dovich effect from clusters of galaxies

By far the commonest references to the Sunyaev-Zel'dovich effect in the literature are to the effect that the atmosphere of a cluster of galaxies has on the CMBR. Cluster atmospheres are usually detected through their X-ray emission, as in the example shown in Fig. 2, although the existence of such gas can also be inferred from its effects on radio source morphologies (e.g., Burns & Balonek 1982) - `disturbed' lobe shapes and head-tail sources being typical indicators of the presence of cluster gas.

If a cluster atmosphere contains gas with electron concentration n_e (r), then the scattering optical depth, Comptonization parameter, and X-ray spectral surface brightness along a particular line of sight are

(61)

(62)

(63)

where z is the redshift of the cluster, and is the spectral emissivity of the gas at observed X-ray energy E or into some bandpass centered on energy E (including both line and continuum processes). The factor of 4 in the expression for b_X arises from the assumption that this emissivity is isotropic, while the (1 + z)³ factor takes account of the cosmological transformations of spectral surface brightness and energy.

By far the most detailed information on the structures of cluster atmospheres is obtained from X-ray astronomy satellites, such as ROSAT and ASCA. Even though these satellites also provide some information about the spectrum of b_X (and hence an emission-weighted measure of the average gas temperature along the line of sight) there is no unique inversion of (63) to n_e (r) and T_e (r). Thus it is not possible to predict accurately the distribution of y on the sky, and hence the shape of the Sunyaev-Zel'dovich effect (which we will, in the current section, take to be close to the shape of y, although Sec. 3.3 indicates that an accurate prediction of the Sunyaev-Zel'dovich effect requires a more complicated calculation which includes both the electron scattering optical depth and the gas temperature).

In many cases it is then convenient to introduce a parameterized model for the properties of the scattering gas in the cluster, and to fit the the values of these parameters to the X-ray data. The integral (62) can then be performed to predict the appearance of the cluster in the Sunyaev-Zel'dovich effect. A form that is convenient, simple, and popular is the isothermal model, where it is assumed that the electron temperature T_e is constant and that the electron number density follows the spherical distribution

(64)

(Cavaliere & Fusco-Femiano 1976, 1978: the so-called `isothermal beta model'). This has been much used to fit the X-ray structures of clusters of galaxies and individual galaxies (see the review of Sarazin 1988). Under these assumptions the cluster will produce circularly-symmetrical patterns of scattering optical depth, Comptonization parameter and X-ray emission, with

(65)

(66)

(67)

where the central values are

(68)

(69)

(70)

is the angle between the center of the cluster and the direction of interest and _c = r_c / D_A is the angular core radius of the cluster as deduced from the X-ray data. D_A is the angular diameter distance of the cluster, given in terms of redshift, deceleration parameter q₀ , and Hubble constant by

(71)

if the cosmological constant is taken to be zero (as it is throughout this review).

A useful variation on this model was introduced by Hughes et al.(1988) on the basis of observations of the Coma cluster. Here the divergence in gas mass which arises for typical values of that fit X-ray images is eliminated by truncating the electron density distribution. A model structure of similar form describes the decrease of gas temperature at large radius. The density and temperature functions used are

(72)

(73)

where r_lim is the limiting gas radius, r_iso is the isothermal radius, and is some index. Not all choices of these parameters are physically reasonable, but the forms above provide an adequate description of at least some cluster structures. Further modifications of models (72 - 73) are required in cases where the cluster displays a cooling flow (Fabian et al. 1984), but this will be important only in Sec. 11.1 in the present review.

For cluster CL 0016+16 shown in Fig. 2, the redshift of 0.5455 implies an angular diameter distance D_A = 760 h₁₀₀^-1 Mpc (for q₀ = 0.5). The X-ray emission mapped with the ROSAT PSPC best matches a circular distribution of form (67) with structural parameters = 0.73 ± 0.02 and _c = 0.69 ± 0.04 arcmin, so that r_c = (150 ± 10)h₁₀₀^-1 kpc. The corresponding cluster central X-ray brightness, b₀ = 0.047 ± 0.002 counts s^-1 arcmin^-2 (Hughes & Birkinshaw 1998). X-ray spectroscopy of the cluster using data from the GIS on the ASCA satellite and the ROSAT PSPC led to a gas temperature k_BT_e = 7.6 ± 0.6 keV and a metal abundance in the cluster that is only 0.07 solar. The X-ray spectrum is absorbed by a line-of-sight column with equivalent neutral hydrogen column density N_H = (5.6 ± 0.4) x 10²⁰ cm^-2. These spectral parameters are consistent with the results obtained by Yamashita (1994) using the ASCA data alone. ⁽⁵⁾

Using the known response of the ROSAT PSPC and the spectral parameters of the X-ray emission found from the ASCA data, the emissivity of the intracluster gas in CL 0016+16 is _e0 = (2.70 ± 0.06) x 10^-13 counts s^-1 cm^-5. The central electron number density can then be found from (70) to be about 1.2 x 10^-2 h₁₀₀^1/2 electrons cm^-3. The corresponding central optical depth through the cluster is _e0 = 0.01 h₁₀₀^-1/2, which corresponds to a central Comptonization parameter y₀ = 1.5 x 10^-4h₁₀₀^-1/2. At such a small optical depth, the Sunyaev-Zel'dovich effect through the cluster should be well described by (51), so that the brightness change through the cluster center should be T₀ = -2y₀T_rad -0.82 h₁₀₀^-1/2 mK at low frequency.

More complicated models of the cluster density and temperature can be handled either analytically, or by numerical integrations. For example, Fig. 2 clearly shows a non-circular structure for CL 0016+16: a better representation of the structure of the atmosphere may then be to replace (64) by an ellipsoidal model, with

(74)

where the matrix M encodes the orientation and relative sizes of the semi-major axes of the cluster. If CL 0016+16 is assumed to be intrinsically oblate, with its symmetry axis in the plane of the sky (at position angle -40°) and with structural parameters that match the X-ray image, then the intrinsic axial ratio is 1.17 ± 0.03, with best-fitting values for and the major axis core radius of 0.751 ± 0.025 and 0.763 ± 0.045 arcmin (Hughes & Birkinshaw 1998). The central X-ray surface brightness is almost unchanged, reflecting the good degree of resolution of the cluster structure effected by the ROSAT PSPC. With these structural parameters, the predicted Sunyaev-Zel'dovich effect map of the cluster is as shown in Fig. 10. The predicted central Sunyaev-Zel'dovich effect is now T₀ -0.84h₁₀₀^-1/2 mK at low frequency, very little changed from the prediction of the circular model.

Figure 10. A model for CL 0016+16 assuming that the cluster is oblate with the symmetry axis in the plane of the sky, and with the structural parameters fixed by fits to the ROSAT PSPC image (Fig. 2). Left: the model X-ray surface brightness. Right: the model Sunyaev-Zel'dovich effect. The contours in both plots are spaced at intervals of 12.5 per cent of the peak effect - note that the Sunyaev-Zel'dovich effect shows a much greater angular extent than the X-ray emission (compare the angular dependences in equations 66 and 67. The central Sunyaev-Zel'dovich effect predicted on the basis of the X-ray data is -0.84h100-1/2 mK.

However, most of the recent interest in the Sunyaev-Zel'dovich effects of clusters has not been because of their use as diagnostics of the cluster atmospheres, but rather because the effects can be used as cosmological probes. A detailed explanation of the method and its limitations is given in Section 11, but the essence of the method is a comparison of the Sunyaev-Zel'dovich effect predicted from the X-ray data with the measured effect. Since the predicted effect is proportional to h₁₀₀^-1/2 via the dependence on the angular diameter distance (equation 68 with r_c = D_{A c}; see also the discussion of CL 0016+16 above), this comparison measures the value of the Hubble constant, and potentially other cosmological parameters.

It should be emphasized that the Sunyaev-Zel'dovich effect has the unusual property of being redshift independent: the effect of a cluster is to cause some fractional change in the brightness of the CMBR, and this fractional change is then seen at all positions on the line of sight through the cluster, at whatever redshift. Thus the central Sunyaev-Zel'dovich effect through a cluster with the properties of CL 0016+16 will have the same value whether the cluster is at redshift 0.55, 0.055, or 5.5. This makes the Sunyaev-Zel'dovich effect exceptionally valuable as a cosmological probe of hot electrons, since it should be detectable at any redshift for which regions with large electron pressures exist.

There have now been a number of detections of the Sunyaev-Zel'dovich effects of clusters, and recent improvements in the sensitivities of interferometers with modest baselines have led to many maps of the effects. A discussion of the detection strategies, and the difficulties involved in comparing the results from different instruments, is given in Section 8.