The Sunyaev-Zeldovich effect involves scattering of CMB
photons by rapidly moving electrons in the hot gas
in clusters of galaxies.

The diagram above shows a cartoon version of this effect. It is possible to use a combination of the Sunyaev-Zeldovich effect and the X-ray emission from the hot gas to derive a distance to the cluster. This effect is proportional to (1) the number density of electrons, (2) the thickness of the cluster along our line of sight, and (3) the electron temperature. The parameter that combines these factors is called the Kompaneets

Many years ago
Sunyaev and Zeldovich (1980, ARAA, 18, 537)
published a figure similar to the one below.
This figure still appears in print, for example as Figure 1 in
Carlstrom, Holder and Reese (2002, ARAA, 40, 643).

I am not clear about the purpose of the arrows in the figure but I have put them in to make my version of the figure more like the Sunyaev and Zeldovich figure. This figure is drawn with a very large value of

Since I got tired of seeing the weird red spectrum in too many talks, I have generated the exact spectrum plot shown below.

I hope that people speaking about the SZ effect and wanting a plot showing both the blackbody and the distorted spectrum on a log-log scale will use the figure above instead of some version of the old inaccurate Figure 4 from S-Z (1980).

For real data from clusters, the values of *y* are much smaller.
A really big value of *y = 0.0005* gives a distorted spectrum
that is hardly distinguishable from the blackbody on a log-log graph.
Therefore one typically plots the difference in intensity between the
on-cluster distorted spectrum and the off-cluster blackbody spectrum.
Such a plot is shown
below with both the exact and the first-order approximation:

Clearly the first-order approximation is perfectly suitable for analyzing real data, since the red and blue curves are very nearly coincident for this value of

For reference, the first order spectrum is *I = B+y*dI/dy* with
*dI/dy* given by

while the exact equation is based on convolving the input spectrum with a Gaussian in log frequency space:

The plots included above were created using Postscript files that evaluated these formulae, such as this Postscript file for the second plot.

One can also worry about relativistic effects when the electron
temperature is high. The S-Z formulae are correct for
*kT << mc ^{2}*.
Wright (1979, ApJ, 232, 348) gives an exact numerical treatment
for a very hot plasma in the context of a hot intergalactic medium
that produces both the X-ray background and a distorted CMB spectrum.
This IGM model was ruled out by the COBE FIRAS data but the formulae
for computing the distortion are still correct.
Fabbri (1981, ApSS, 77, 529) was able to replace some of the
numerical integrations in Wright (1979) by analytic formulae.
Itoh and Nozawa (2004, A&A, 417, 827) provide tables
of the relativistic corrections and an accurate analytic fitting
formula.
These corrections are not too large for typical clusters with

The figure above shows the change in intensity for

Tutorial:
Part 1 |
Part 2 |
Part 3 |
Part 4

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© 2007-2008 Edward L. Wright. Last modified 20 Jan2008