We present the first fully and inherently relativistic derivation of the thermal Sunyaev-Zel'dovich effect. This work uses the formalism historically applied to compute radiation spectra emerging from inverse Thomson/Compton sources of x-ray radiation. Comparing our results to the traditional approach based on relativistically-corrected classical Kompaneets equation, we find small, but systematic differences. Most notable are the modest (⩽10%) differences in the crossover frequency where the spectral distortion due to the Sunyaev-Zel'dovich effect vanishes, and the quadratic scaling of the energy shift with the electron cloud temperatures.

中文翻译：

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热 Sunyaev-Zel'dovich 效应的完全相对论推导


我们提出了热 Sunyaev-Zel'dovich 效应的第一个完全和本质相对论推导。这项工作使用历史上应用于计算 X 射线辐射逆汤姆逊/康普顿源产生的辐射光谱的形式。将我们的结果与基于相对论校正的经典 Kompaneets 方程的传统方法进行比较，我们发现了微小但系统性的差异。最值得注意的是交越频率的适度 （⩽10%） 差异，其中由 Sunyaev-Zel'dovich 效应引起的光谱失真消失，以及能量偏移随电子云温度的二次缩放。