5 Current and future CMB 4 Observing modes and foregrounds4.1 The SZ effect

4.2 Relativistic corrections to the SZ effect

The above treatment of the SZ effect is purely non-relativistic. In clusters where tex2html_wrap_inline1743 keV, relativistic effects become important. The relativistic effect can be included either by extending the equations to include relativistic terms ([41], [94], [95]) or by including multiple scattering descriptions of the Comptonization process ([100], [49], [96], [71], [84]). Both of these approaches give consistent results. To first order in temperature the correction for the thermal SZ effect is given by

displaymath227

  equation239

and in the Rayleigh-Jeans limit (small x) we find

  equation252

and so it is seen that the inclusion of the relativistic treatment tends to lead to a small decrease in the SZ effect. The Hubble constant is inferred from combined Sunyaev-Zel'dovich and X-ray data by a relation of the form [66]

  equation261

The reduction in tex2html_wrap_inline1527 in the Rayleigh-Jeans region, for given cluster parameters, leads to a decrease in the constant of proportionality in Equation  12Popup Equation, and hence a small reduction in the determined values of tex2html_wrap_inline1567 . For example, it is found that if the cluster temperature is 8keV the reduction in tex2html_wrap_inline1567 due to relativistic effects is 5%.



5 Current and future CMB 4 Observing modes and foregrounds4.1 The SZ effect

image The Cosmic Microwave Background
Aled W. Jones and Anthony N. Lasenby
http://www.livingreviews.org/lrr-1998-11
© Max-Planck-Gesellschaft. ISSN 1433-8351
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