## Thermal Sunyaev-Zeldovich Effect

The thermal Sunyaev-Zeldovich effect arises from the frequency shift when CMB photons are scattered by the hot electrons in the intra-cluster gas. The frequency dependence of this effect (see Figure 1.14), results in a temperature decrement in the Rayleigh-Jeans region of the CMB spectrum and to a temperature excess at high frequencies. The central frequencies of the PLANCK bolometer bands have been carefully chosen to straddle the regions of negative and positive decrement, with one channel centered at 217 GHz, where the thermal SZ flux is zero. This arrangement has been chosen to optimize the separation of the thermal SZ effect from the frequency independent primary anisotropy pattern.

The amplitude of the SZ effect in a particular direction can be characterized by the Compton parameter , where is the Thomson cross-section and the integral is taken along the line-of-sight. The net SZ flux from a cluster can therefore be written

 (11)

where describes the frequency dependence of the Compton distortion (see figure 1.14.a), A is a normalization constant and , where the integral extends over the solid angle subtended by the cluster. The integral Y is proportional to the total gas mass within the cluster times the mass-weighted temperature.

Observations of the SZ effect provide information on the hot intra-cluster gas that is complementary to that derived from observations at X-ray wavelengths. A key difference between the SZ and X-ray fluxes of clusters arises from their different scaling with the electron density (ne2 for X-rays, and ne for SZ): while the X-ray emission is strongly peaked near the centre, the signal-to-noise in the SZ signal remains roughly constant in logarithmic rings around the centre. This remains true insofar as the gas is approximately isothermal (i.e. within the virialized region of the cluster) defining a characteristic radius , where rc is a typical cluster core radius.

##### Figure 1.15: Profiles of a 4 keV cluster in X-ray brightness (solid, with scale on the left axis in counts in 0.3-10 keV for the XMM-EPIC instrument) and in the Compton y parameter (dashes with scale on the right axis) for clusters with a King-type profile, a core radius rc = 0.3 Mpc, and a central electron density ne0 = 2 10-2cm-3. This Abell 496-like cluster is viewed at a redshift z = 0.1 (left) and z = 0.2 with in addition a cut-off for the virialized gas at 20rc (right). The horizontal line shows the level of the diffuse X-ray background.

Figure 1.15 shows the differences between the X-ray and y profiles of a typical rich cluster of galaxies at two different redshifts. The scales correspond to typical sensitivities for an X-ray mission such as XMM and for the PLANCK mission.

The net SZ flux from a cluster is insensitive to cluster redshift, due to the increase of the CMB temperature with z. Hence we expect distant clusters () to be observable with PLANCK, although these will not be resolved. Figure 1.14.b shows the ratio of the expected X-ray brightness to the SZ brightness at mm, which decreases strongly with z for different central temperatures of cluster. The main effect is the z dependence: the ratio has decreased by a factor of 30 at z=1. Therefore, the SZ effect provides a powerful tool with which to study the evolution of clusters, as described in further detail below. The sensitivity of PLANCK to high redshift clusters can be utilized to complement X-ray and optical/near-IR investigations. For example, high z clusters detected by PLANCK can be selected as targets for XMM observations and for ground based/HST spectroscopy and imaging.

Observations of the SZ effect with PLANCK will circumvent a further problem of X-ray observations. The peaked character of the X-ray emissivity allows gas mass determinations in a very limited radius, of a few rc only (except for very long integrations possible only on a small sample of clusters). However, PLANCK observations will probe the gas properties far beyond a few core radii. This is illustrated in figure 1.15 which shows X-ray brightness and y profiles for typical rich clusters at two different redshifts. The PLANCK sensitivity allows the detection of y values of within rings of order one degree (see From Observations to Scientific Information). The SZ effect is thus sensitive enough to observe the temperature drop which defines the limit of virialization.

The EPIC instrument, on the XMM observatory, in a 20-hour observation, will be able to map the same cluster up to about . This shows that the most powerful strategy for learning about the properties and evolution of the gas in clusters will be to combine PLANCK observations with X-ray brightness profiles and X-ray temperature measurements.

The combination of spatially resolved X-ray temperature and flux profiles, and measurements of the thermal SZ effect in the CMB, can be used to estimate the true spatial dimensions of rich clusters of galaxies and hence to estimate the Hubble constant (Gunn 1978, Silk & White 1978). Ground based SZ measurements already provide useful constraints on the Hubble constant, suggesting a value of (Rephaeli 1995). PLANCK will produce spatially resolved thermal SZ maps for nearby rich clusters which will be complementary to those of purpose-designed ground based experiments such as the submillimetre SUZIE experiment (Wilbanks et al. 1994), covering a wider spectral range and with better control of systematic sources of error. PLANCK SZ measurements of the many thousands of more distant, unresolved, clusters can also be used to estimate H0 and to constrain the deceleration parameter q0. The main limitation in this case is likely to arise from the difficulty of the X-ray observations, which must have good spatial resolution, rather than from the CMB measurements where all that is required is a high signal-to-noise measurement of the integrated SZ effect.

##### Figure 1.16: (Left) Differential number counts versus redshift for different flux limits and . (Right) Number counts per square degree versus flux limit for two values of .

To evaluate the number of clusters detectable by PLANCK, we refer again to our simulations which have shown (see From Observations to Scientific Information) that PLANCK will detect clusters with when they cover more than one pixel (70% detection rate for a one year mission), or if rc subtends less than . The integrated value is , where y0 stands for the central value and is expressed in arcmin.

The exact number of clusters detectable through the SZ effect, as a function of z, depends on

• the cosmological parameters,
• the shape of the present mass and temperature distributions of galaxy clusters, and
• the cosmological evolution of the mass and temperature distributions (which in turn depends on the model for galaxy and structure formation).

Analytic estimates based on the Press-Schechter (1974) mass function suggest the presence of about 15 000 such clusters over the whole sky (assuming ), or up to 3 times higher for lower values of (figure 1.16.b). This dependence on is a consequence of the earlier formation of clusters of galaxies in a low density universe. Figure 1.16.a shows the corresponding redshift distribution. Since these estimations are strongly model dependent, measurements of the SZ effect will provide sensitive tests of theories of structure formation.

In summary, PLANCK will detect many thousands of rich clusters of galaxies via the SZ effect, and significant numbers are expected to lie at redshifts z > 0.2 and perhaps as high as . The observed counts of clusters will provide a powerful test of models of structure formation and evolution. Clusters observed by PLANCK can be selected as targets for X-ray observations with new satellites such as XMM, to study the properties and evolution of the intracluster gas and as targets for optical and near-infrared observations to study galaxy evolution and gravitational lensing.