### 4.2 Relativistic corrections to the SZ effect

The above treatment of the SZ effect is purely non-relativistic. In clusters where ,
relativistic effects become important. The relativistic effect can be included either by extending
the equations to include relativistic terms ([17], [88], [89]) or by including multiple scattering
descriptions of the Comptonization process ([100], [29], [91], [54], [76]). Both of these approaches give
consistent results. To first order in temperature the correction for the thermal SZ effect is given
by

and in the Rayleigh–Jeans limit (small ) we find
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 [49]
The reduction in in the Rayleigh–Jeans region, for given cluster parameters, leads to a decrease in the
constant of proportionality in Equation 12, and hence a small reduction in the determined values of .
For example, it is found that if the cluster temperature is 8 keV the reduction in due to relativistic
effects is 5%.