I can't find any convincing answer for the following question :

Why do we always (or often) plot the CMB power spectrum in this way?

Graph of CMB multipole moment

I mean the vertical axis is $C_\ell \ell (\ell+1)$ and not only $C_\ell$. Why?

I know it's because of the scale invariance, but why do we absolutely want to show the flat line at low $\ell$? And I do not understand why the power spectrum is flat in this scale.


2 Answers 2


I haven't got a great answer for this, but since no-one else has answered ...

As you mention, for the Sachs-Wolfe effect the $C_{\ell}$ values drop off as approximately $\ell(\ell + 1)$ so plotting $C_{\ell}\ell(\ell + 1)$ on the $y$ axis gives an approximately horizontal line and this makes it easy to see deviations from Sachs-Wolfe behaviour. However I suspect the main reason the graphs are drawn this way is that it nicely highlights the doppler peaks. If you just plotted $C_{\ell}$ you'd need to use a log axis and that would make all the peaks look smaller.

  • $\begingroup$ Thanks for your answer. I think you're quite close to the aim of plotting the CMB PS in this way. $\endgroup$
    – AnSy
    May 19, 2012 at 8:40
  • 4
    $\begingroup$ Well, because we're really plotting the anisotropy i.e. variations. So they're the Fourier modes not of the temperature $T$ itself but its Laplacian over the sphere, $\Delta T$, and the Laplacian has a simple well-defined effect on the component $C_l$ which is multiplied by a spherical harmonic function $Y_{lm}$: it just multiplies the spherical harmonic function by $-l(l+1)$. That's why $l(l+1)$ may be identified with the (minus) Laplacian. It's more natural to insert the Laplacian rather than not to really measure "variations" and to get rid of the huge constant term prop. to $Y_{00}$, too. $\endgroup$ May 19, 2012 at 8:48

I learned in my class on cosmology that

$C_{\ell}\ell(\ell + 1) \propto (\Delta T)^2$

but don't have any sources to back this up, besides, the slides from my professor. See slide 17 and 18 of this talk https://neutrino.ikp.kit.edu/personal/drexlin/data/_uploaded/file/Kosmo1/CS08.pdf , which is unfortunately in german, but the equations should be understandable anyway.


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