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

42 ( x) 7 ( x) 7x2 ( ( x) )) ] − ---x2coth2 -- + ---x3coth3 -- + -------(--) x coth -- − 3 ] (10 ) 5 2 10 2 5 sinh2 x2 2
and in the Rayleigh–Jeans limit (small x) we find
⌊ ( )⌋ 17 kT ( kT )2 (ΔTRJ )thermal = − 2T0y⌈1 − -----e-+ O ( --e- )⌉ (11 ) 10 me me
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]
H ∘ ∝ ΔT − 2. (12 )
The reduction in ΔT in the Rayleigh–Jeans region, for given cluster parameters, leads to a decrease in the constant of proportionality in Equation 12View Equation, 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%.


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