The statistical nature of the second order corrections to the thermal SZE

This paper shows that the accepted expressions for the second order corrections in the parameter z to the thermal Sunyaev-Zel’dovich effect can be accurately reproduced by a simple convolution integral approach. This representation allows to separate the second order SZE corrections into two type of components. One associated to a single line broadening, directly related to the even derivative terms present in the distortion intensity curve, while the other is related to a frequency shift, which is in turn related to the first derivative term. The thermal Sunyaev-Zel’dovich effect (SZE) [1] [2] arises from the frequency shift of CMB photons that are scattered by a hot electron gas. It has been detected in observations of some rich, X-ray luminous clusters. The SZE can be applied to the determination of cosmological parameters [3] [4]. The distorted spectra arising from Compton scattering can be expressed as a power series containing terms which are even powers of the frequency ν, except for a first order term in ν that is associated to the photon diffusion approximation [5] [6] [7] [8]. The parameter z = kTe mc2 used as a discriminant to indicate when the relativistic corrections are important is usually kept up to second order effects. In this paper we show that the scattering law approach to the SZE [9] is able to reproduce the distortion curve contained in Refs. [5] [6] [7] [8] [10] providing new insight of the physics inherent in distortion spectrum. The basic idea is simple. The distorted spectrum can be obtained by performing a convolution integral between the CMB blackbody curve and a kernel corresponding to a single line profile, slightly