# CMB parameter, what's the meaning of the matter power spectrum normalization $\sigma_8$?

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 $\sigma_8$?

• Most important question: how does WMAP/Planck calculate values for $\sigma_8$? – ShanZhengYang Feb 25 '15 at 14:28
• This link section 1.3.5 may be helpful. arxiv.org/pdf/1401.1389.pdf Also, this link arxiv.org/abs/1303.5080 – yess Sep 24 '15 at 12:54
• @yess That's the clearest explanation I've read. You should answer below so I can close this. – ShanZhengYang Oct 1 '15 at 9:37
• 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. – Godfrey Miller Oct 2 '15 at 2:59
• @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 – ShanZhengYang Oct 2 '15 at 15:11