Relativistic corrections to 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 $kBTe > 10 keV$ , 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

( ) ( ) [---x2ex-- x- kTe( 47- x- (ΔTRJ )thermal = T0y (ex − 1)2[x coth 2 − 4 + me − 10 + 2 xcoth 2

and in the Rayleigh–Jeans limit (small $x$ ) 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 $ΔT$ 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 $H ∘$ . For example, it is found that if the cluster temperature is 8 keV the reduction in $H ∘$ due to relativistic effects is 5%.