= Solution
The <thermal Sunyaev-Zeldovich effect> is inverse-Compton scattering of <cosmic microwave background> <photons> by hot <electrons> in the <intracluster medium>. <Electron> thermal <energy> shifts <photons> toward higher frequencies, changing the spectrum relative to neighboring sky. In the nonrelativistic, small-optical-depth limit its amplitude is controlled by the Compton parameter
$$
\boxed{y=\frac{\sigma_T}{m_ec^2}\int n_e k_BT_e\,dl.}
$$
It measures line-of-sight <electron> <pressure>. At low frequencies the apparent Rayleigh-Jeans <temperature> changes by $\Delta T/T_{\rm CMB}\simeq-2y$. The thermal spectral distortion crosses through zero near $217\,\mathrm{GHz}$ for the usual CMB <temperature> and then becomes an increment; relativistic corrections matter for hot gas. Multi-frequency measurements distinguish this spectrum from dust, radio sources and an ordinary CMB <temperature> fluctuation.
The surface-brightness contrast has no ordinary $(1+z)^{-4}$ dimming for the same cluster <pressure> integral: the incident CMB <temperature> and the observation's <frequency> redshift scale together. Angular extent and integrated flux still depend on distance. This makes the effect useful for selecting distant clusters and mapping their thermal <pressure>. The solid-angle integral $Y=\int y\,d\Omega$ scales with total thermal <energy> divided by angular-diameter distance squared, providing a <mass> proxy when its gas-physics calibration is specified.<Planck 2013 results. XXIX. Planck catalogue of Sunyaev-Zeldovich sources by Planck Collaboration>[cluster survey and integrated-SZ selection context]{ref}
X-ray surface brightness depends roughly on $\int n_e^2\Lambda(T_e)dl$, whereas $y$ depends on $\int n_eT_e dl$. Combining X-ray spectra, brightness and resolved SZ data constrains density and line-of-sight depth. With a justified geometry, comparison to angular size gives a distance estimate; clumping and elongation can bias it. Cluster abundances and their redshift evolution test cosmological structure growth after selection and <mass> calibration are modeled.
The <kinetic Sunyaev-Zeldovich effect> additionally probes bulk line-of-sight motion, with leading $\Delta T/T_{\rm CMB}\simeq-\tau_e v_\parallel/c$ in a stated velocity-sign convention. Its approximately blackbody <temperature> spectrum differs from the thermal distortion. These are complementary measurements of thermal <pressure> and gas motion, requiring foreground and instrumental control.
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