Astronomy and astrophysics Radiation processes and radiative transfer Proposition

Sunyaev–Zeldovich effect

Builds on Kompaneets equation Inverse Compton scattering Black body

Statement

Let the cosmic microwave background be black-body radiation of temperature $T_0$ , with occupation number $n_0(x) = 1/(e^x - 1)$ . Here $x = h\nu/(k_B T_0)$ , $\nu$ is the frequency, $h$ the Planck constant and $k_B$ the Boltzmann constant. Let this radiation cross a cloud of free electrons at rest relative to it, with number density $n_e$ and temperature $T_e$ . Assume $k_B T_0 \ll k_B T_e \ll m_e c^2$ , where $m_e$ is the electron mass and $c$ the speed of light. Let $\sigma_T$ be the Thomson cross-section and $\ell$ the length along the line of sight. Assume the Compton parameter is small: $y = \int \frac{k_B T_e}{m_e c^2} \, n_e \sigma_T \, d\ell \ll 1.$ Then to first order in $y$ the occupation number of the emerging radiation is $n_0 + \Delta n$ with $\Delta n(x) = y \, \frac{x e^x}{(e^x - 1)^2} \left( x \coth\frac{x}{2} - 4 \right).$ Equivalently, the change of the temperature of the black body that has the same occupation number at frequency $\nu$ is $\Delta T / T_0 = y \, (x \coth(x/2) - 4)$ .

Remarks

The proof applies the Kompaneets equation for a short time. Since $T_e \gg T_0$ , the terms $n + n^2$ are negligible against $\partial n / \partial x$ in the variable $h\nu/(k_B T_e)$ , and the equation becomes $\Delta n = y \, x^{-2} \partial_x (x^4 \partial_x n_0)$ in the variable $x$ . A direct computation with $n_0 = 1/(e^x - 1)$ gives the formula. At low frequencies, $x \ll 1$ , one has $x \coth(x/2) \to 2$ , so $\Delta T / T_0 \approx -2y$ : the cloud appears as a cold spot. The change vanishes at $x \approx 3.83$ , about $217$ GHz, and is positive above it; this spectral shape distinguishes the effect from other temperature fluctuations. The formula assumes non-relativistic electrons; for very hot clusters relativistic corrections are needed.

Going deeper

For a student

This is the thermal Sunyaev–Zeldovich effect; a cloud moving relative to the background gives an additional kinematic effect. The signal depends on $y$ , which is proportional to the integrated electron pressure along the line of sight, and not on the distance of the cluster. So surveys of the effect find clusters out to large redshift. It follows from the Kompaneets equation, which is the diffusion limit of inverse Compton scattering.

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Tag	`D4J2Z7J7K2`
URI	`m2://D4J2Z7J7K2`
Kind	statement, proposition
Writing	W0 stub
MSC	85A25, 85A40
Status	This copy of the site has no build data. Run `m2 status D4J2Z7J7K2`.
About	Kompaneets equation, Inverse Compton scattering
Uses	Kompaneets equation, Inverse Compton scattering, Black body
Same as	Wikidata Q1519204 Wikipedia Sunyaev–Zeldovich effect
Changed	3 November 2026

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