= Solution
The <thermal Sunyaev-Zeldovich effect> is <inverse Compton scattering> of <cosmic microwave background> <photons> by hot cluster <electrons>. For an <optically thin>, nonrelativistic thermal <plasma>,
$$
y=\frac{\sigma_T}{m_ec^2}\int n_ek_BT_e\,dl,\qquad
\Delta I_\nu=I_0y\frac{x^4e^x}{(e^x-1)^2}\left[x\coth(x/2)-4\right],
\quad x=\frac{h\nu}{k_BT_{\rm CMB}},\quad I_0=\frac{2(k_BT_{\rm CMB})^3}{h^2c^2}.
$$
The <Compton y parameter> measures the <electron> <pressure> column. The distortion is a decrement below roughly \b[217 GHz], an increment above it, and has $\Delta T/T=-2y$ in the <Rayleigh-Jeans law> limit. Relativistic <electron> corrections modify the spectrum and null.<Relativistic corrections to the Sunyaev-Zel'dovich effect by A. D. Challinor and A. N. Lasenby>[abstract and relativistic thermal-distortion calculation]{ref}
Multifrequency maps separate this spectrum from primary <Cosmic microwave background anisotropy>, radio sources and dust; atmospheric subtraction, beams and <calibration> require control. The <kinetic Sunyaev-Zeldovich effect> has $\Delta T/T\simeq-\tau v_{\rm los}/c$ and a CMB-temperature-like spectrum, so it is both a contaminant and a peculiar-velocity probe. Planck demonstrated blind multifrequency cluster detection with external confirmation and a measured selection function.<Planck 2013 results. XXIX. Planck catalogue of Sunyaev-Zeldovich sources by Planck Collaboration>[abstract; multifrequency detection and catalogue validation]{ref}
For a fixed physical <pressure> column, $y$ has \b[no ordinary <cosmological surface-brightness dimming>]: the distorted spectrum and CMB redshift together. The <integrated Compton parameter> $Y=\int y\,d\Omega$ still scales as $D_A^{-2}$ for fixed integrated physical <pressure>. SZ observations therefore find distant clusters and measure thermal <pressure>; integrated $Y$ is a useful mass proxy, requiring <calibration> and a selection model for cosmological counts. X-ray emission scales approximately as $\int n_e^2\Lambda(T_e)dl$, while SZ scales as $\int n_eT_e dl$. Combining them and a measured <temperature> constrains <density>, depth and, with angular geometry, distance. <gas clumping factor>, asphericity, <temperature> structure and nonthermal support limit that inference.
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