Most CMB experiments like WMAP and Planck include a certain cosmological parameter called $\sigma_8$. My understanding is that normalization of the matter power spectrum is not a theoretical prediction, but rather must be normalized by observation. So, the normalization is parameterized by this $\sigma_8$, the linear theory amplitude of matter fluctuations on 8 $h^{-1}$ Mpc scales.

How is quantity derived? Why was it chosen? Where does it come from?

How is this actually measured by WMAP and Planck, i.e. how does the angular power spectrum $C_l$ explain this quantity? How does the data give values for $\sigma_8$?

  • $\begingroup$ Most important question: how does WMAP/Planck calculate values for $\sigma_8$? $\endgroup$ Feb 25, 2015 at 14:28
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    $\begingroup$ This link section 1.3.5 may be helpful. arxiv.org/pdf/1401.1389.pdf Also, this link arxiv.org/abs/1303.5080 $\endgroup$
    – yess
    Sep 24, 2015 at 12:54
  • $\begingroup$ @yess That's the clearest explanation I've read. You should answer below so I can close this. $\endgroup$ Oct 1, 2015 at 9:37
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    $\begingroup$ For a theoretical overview, I recommend Chapter 8 (The Growth of Structure) of Cosmology by Steven Weinberg. Section 8.1 (Linear perturbations after recombination) contains a brief discussion of the empirical normalization of the matter power spectrum. $\endgroup$ Oct 2, 2015 at 2:59
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    $\begingroup$ @GodfreyMiller Yes, in Chapter 8 he introduces the power spectral function P(k) and explains how large surveys measure P(k) from the angular average of the square of a Fourier integral of the matter density perturbation over the survey volume. Then, in equation (8.1.43) Weinberg shows how knowledge of the power spectral function allows us to calculate the mean square value of the fractional density fluctuation, /sigma^2. That's the theoretical meaning. I was looking for some heavy details as to how Planck calculated this value, but I now see this above. Thanks $\endgroup$ Oct 2, 2015 at 15:11

1 Answer 1


Transferring from the comments, as they do have a tendency to disappear:

There are two useful references given by @yess, one a Planck publication, the other a comprehensive report on the cosmological parameters.

From the latter is is evident that sigma_8 is derived from other fitted to data parameters .

c1 c2 c3

One should read the references for a comprehensive view.

  • $\begingroup$ Both your links point to the same arXiv paper. Could you correct one of them please? A reviewed article would be more reliable than unreviewed... $\endgroup$
    – FJC
    Oct 19, 2016 at 16:21
  • $\begingroup$ @FJC thanks for catching it, although the reference was still there in the comment of yess. I updated the second one too. $\endgroup$
    – anna v
    Oct 19, 2016 at 18:01

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