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].