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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.

Qmechanic
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AnSy
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2 Answers2

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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.

John Rennie
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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.

jak
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