(kinetic SZ-effect).

The relative change $\Delta T/T$ in thermodynamic CMB temperature at position $\bmath{\theta}$ as a function of dimensionless frequency $x=h\nu /(k_B T_\mathrm{CMB})$ due to the thermal SZ-effect is given by: $$\begin{aligned}
\frac{\Delta T}{T}(\bmath{\theta}) & = & y(\bmath{\theta})\,\left(x\frac{e^x+1}{e^x-1}-4\right)\mbox{ with }\\
y(\bmath{\theta}) & = & \frac{\sigma_\mathrm{T} k_B}{m_{\mathrm{e}}c^2}\int{\mathrm{d}}l\:n_{\mathrm{e}}(\bmath{\theta},l)T_{\mathrm{e}}(\bmath{\theta},l)\mbox{,}
\label{sz_temp_decr}\end{aligned}$$ where the amplitude $y$ of the thermal SZ-effect is commonly known as the thermal Comptonisation parameter, that itself is defined as the line-of-sight integral of the temperature weighted thermal electron density. $m_{\mathrm{e}}$, $c$, $k_B$ and $\sigma_\mathrm{T}$ denote electron mass, speed of light, Boltzmann’s constant and the Thompson cross section, respectively. The kinetic SZ-effect arises due to the motion of the cluster parallel to the line of sight relative to the CMB-frame: $$\frac{\Delta T}{T}(\bmath{\theta}) = -w(\bmath{\theta})\mbox{ with }
w(\bmath{\theta}) = \frac{\sigma_\mathrm{T}}{c}\int{\mathrm{d}}l\:n_{\mathrm{e}}(\bmath{\theta,l})\upsilon_r(\bmath{\theta},l)\mbox{.}$$ Here, $\upsilon_r$ is the radial component of the cluster’s velocity. The convention is such that $\upsilon_r<0$, if the cluster is moving towards the observer. In this case, the CMB temperature is increased. In analogy, the quantity $w$ is refered toas the kinetic Comptonisation. The SZ-observables are the line-of-sight Comptonisations integrated over the solid angle subtended by the cluster. The quantities $\mathcal{Y}$ and $\mathcal{W}$ are refered to as the integrated thermal and kinetic Comptonisations, 