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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 parameterIt measures line-of-sight electron pressure. At low frequencies the apparent Rayleigh-Jeans temperature changes by $\Delta T/T_{\mathrm{CMB}}\simeq -2y$. 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 $Y=\int yd\Omega$ scales with total thermal energy divided by angular-diameter distance squared, providing a mass proxy when its gas-physics calibration is specified.^[ref]

X-ray surface brightness depends roughly on $\int n_e^2\Lambda (T_e)dl$, whereas $y$ depends on $\int n_e{T}_{e}dl$. 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 $\Delta T/T_{\mathrm{CMB}}\simeq -{\tau }_e{v}_{\parallel }/c$ 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.

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,The Compton y parameter measures the electron pressure column. The distortion is a decrement below roughly 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.^[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 $\Delta T/T\simeq -\tau v_{\mathrm{los}}/c$ 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.^[ref]

For a fixed physical pressure column, $y$ has no ordinary cosmological surface-brightness dimming: the distorted spectrum and CMB redshift together. The integrated Compton parameter $Y=\int yd\Omega$ 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 $\int n_e{T}_{e}dl$. 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.

Three distinct low-redshift nonlinear contributions are:

Gravitational lensing supplies another nonlinear modification by remapping pre-existing temperature and polarization patterns. It preserves the local blackbody spectrum; it does not generate temperature differences by lensing a perfectly isotropic background.

Write $n=n_0+y\mathcal{L}{n}_0+O(y^2)$, with $n_0=(e^x-1{)}^{-1}$ and $\mathcal{L}=x^{-2}{\partial }_x(x^4{\partial }_x)$. Differentiating the Planck photon distribution givesDividing by $n_0$ proves the first-order thermal Sunyaev-Zeldovich effect:In the Rayleigh-Jeans law limit, $x{e}^x/(e^x-1)\to 1$ and $x\coth (x/2)\to 2$, so $\Delta n/n_0\to -2y$. In the Wien approximation, the prefactor approaches $x$ and the bracket approaches $x-4$, so $\Delta n/n_0\sim yx(x-4)\sim y{x}^2$. The last limit is an asymptotic description of the first-order coefficient: a perturbative prediction at large $x$ still requires $y{x}^2\ll 1$.

Hot electrons transfer energy to the cosmic microwave background, removing low-frequency photons and enhancing the high-frequency tail. The first-order distortion crosses zero where $x\coth (x/2)=4$, at $x\simeq 3.830$; for today's CMB this is near $217\mathrm{GHz}$. Photon conservation follows directly from the boundary termIn contrast, integration by parts gives $\int x^3\Delta ndx=4y\int x^3{n}_{0}dx$, so radiation gains a fraction $4y$ of its original energy. The thermal Sunyaev-Zeldovich effect is thus a spectral distortion, not simply a new blackbody temperature.

A galaxy cluster with bulk peculiar velocity along the line of sight additionally produces the kinetic Sunyaev-Zeldovich effect. Define $v_{\parallel }>0$ for recession and $\tau =\int n_e{\sigma }_{T}d\ell$. To first order in velocity and optical depth, $\Delta T/T=-\tau v_{\parallel }/c$: receding gas gives a decrement and approaching gas an increment. Its occupation number perturbation is $(\Delta T/T)x{e}^x/(e^x-1{)}^2$, the spectrum of a small blackbody temperature change. It remains nonzero at the thermal null and therefore complicates measurements near that frequency.

There is a genuine flux typo in the original PDF. A received bolometric radiative flux must have inverse-square luminosity distance, so the physically consistent thermal bremsstrahlung expression is $S_X\propto d_L^{-2}\int n_e^2{T}_e^{1/2}dV$. The printed single power of $d_L$ cannot imply the requested Hubble constant scaling.

For an approximately uniform, spherical galaxy cluster let its depth be $\ell =q\theta d_A$, where $q$ is a fixed geometrical factor. With fixed profile coefficients $A,B$, the Rayleigh-Jeans law limit of the thermal Sunyaev-Zeldovich effect and the bolometric X-ray radiative flux giveEliminate the electron number density and use $d_L=(1+z{)}^2{d}_A$:The cluster distance from Sunyaev-Zeldovich and X-ray signals is consequentlyAt low cosmological redshift, $d_A\simeq cz/H_0$, sowhen the other observables and profile parameters are held fixed. With the PDF's literal $d_L^{-1}$ instead, $S_X/D^2\propto \ell /(T_e^{3/2}{d}_L)=q\theta /[T_e^{3/2}(1+z{)}^2]$, independent of distance: this explicitly shows why the correction is necessary.

Real galaxy clusters are neither uniform nor generally spherical. The inferred depth depends on the ratio of line of sight depth to transverse extent; elongation and orientation selection therefore bias the distance. density clumps enhance X-ray emission, which weights $n_e^2$, more than the thermal Sunyaev-Zeldovich effect, which weights $n_e$. An unmodelled gas clumping factor greater than one makes the inferred distance too small and the inferred Hubble constant too large in this simple model. Temperature gradients and multiple thermal components bias the X-ray spectral temperature relative to the pressure-weighted Compton y parameter. Metals alter the cooling spectrum, and finite X-ray bands require a spectral and redshift correction to the bolometric approximation. Relativistic thermal corrections, the kinetic Sunyaev-Zeldovich effect, radio sources and primary Cosmic microwave background anisotropy contaminate the signal. Finally, calibration, angular-profile fitting and low-redshift peculiar velocities affect the observables and the relation between measured redshift and expansion distance.