Internal motion of the gas in dark matter halos also give rise to Doppler shifts in the CMB photons. As in the linear Doppler effect, shifts that are first order in the velocity are canceled as photons scatter off of electrons moving in different directions. At second order in the velocity, there is a residual effect. For clusters of galaxies where the temperature of the gas can reach keV, the thermal motions are a substantial fraction of the speed of light . The second order effect represents a net transfer of energy between the hot electron gas and the cooler CMB and leaves a spectral distortion in the CMB where photons on the Rayleigh-Jeans side are transferred to the Wien tail. This effect is called the thermal Sunyaev-Zel'dovich (SZ) effect [Sunyaev & Zel'dovich, 1972]. Because the net effect is of order , it is a probe of the gas pressure. Like all CMB effects, once imprinted, distortions relative to the redshifting background temperature remain unaffected by cosmological dimming, so one might hope to find clusters at high redshift using the SZ effect. However, the main effect comes from the most massive clusters because of the strong temperature weighting and these have formed only recently in the standard cosmological model.
Great strides have recently been made in observing the SZ effect in individual clusters, following pioneering attempts that spanned two decades [Birkinshaw, 1999]. The theoretical basis has remained largely unchanged save for small relativistic corrections as approches unity. Both developements are comprehensively reviewed in [Carlstrom et al, 2001]. Here we instead consider its implications as a source of secondary anisotropies.
The SZ effect from clusters provides the most substantial contribution to temperature anisotropies beyond the damping tail. On scales much larger than an arcminute where clusters are unresolved, contributions to the power spectrum appear as uncorrelated shot noise ( const. or ). The additional contribution due to the spatial correlation of clusters turns out to be almost negligible in comparison due to the rarity of clusters [Komatsu & Kitayama, 1999]. Below this scale, contributions turn over as the clusters become resolved. Though there has been much recent progress in simulations [Refregier et al, 2000,Seljak et al, 2001,Springel et al, 2001] dynamic range still presents a serious limitation.
Much recent work has been devoted to semi-analytic modeling following the technique of [Cole & Kaiser, 1988], where the SZ correlations are described in terms of the pressure profiles of clusters, their abundance and their spatial correlations [now commonly referred to an application of the ``halo model'' see [Komatsu & Kitayama, 1999,Atrio-Barandela & Mücket, 1999,Cooray, 2001,Komatsu & Seljak, 2001]]. We show the predictions of a simplified version in Plate 5b, where the pressure profile is approximated by the dark matter haloprofile and the virial temperature of halo. While this treatment is comparatively crude, the inaccuracies that result are dwarfed by ``missing physics'' in both the simulations and more sophisticated modelling, e.g. the non-gravitational sources and sinks of energy that change the temperature and density profile of the cluster, often modeled as a uniform ``preheating'' of the intercluster medium [Holder & Carlstrom, 2001].
Although the SZ effect is expected to dominate the power spectrum of secondary anisotropies, it does not necessarily make the other secondaries unmeasurable or contaminate the acoustic peaks. Its distinct frequency signature can be used to isolate it from other secondaries (see e.g. [Cooray et al, 2000]). Additionally, it mainly comes from massive clusters which are intrinsically rare. Hence contributions to the power spectrum are non-Gaussian and concentrated in rare, spatially localized regions. Removal of regions identified as clusters through X-rays and optical surveys or ultimately high resolution CMB maps themselves can greatly reduce contributions at large angular scales where they are unresolved [Persi et al, 1995,Komatsu & Kitayama, 1999].