Vertical axis of the CMB power spectrum

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Discussion Overview

The discussion centers on the choice of vertical axis in the Cosmic Microwave Background (CMB) power spectrum, specifically why it is typically represented as ##l(l+1)C_l/2\pi## rather than simply ##C_l##. The scope includes theoretical considerations and references to literature in cosmology.

Discussion Character

  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants note that the representation as ##l(l+1)C_l/2\pi## is often justified by references such as "Modern Cosmology" by Scott Dodelson and an article by Scott and Smoot.
  • One participant suggests that if the CMB power spectrum were a scale-invariant Gaussian random field, the power spectrum would appear flat when plotted with the vertical axis as ##l(l+1)C_l/2\pi##, attributing this to the nature of spherical harmonics not being strictly flat plane waves.
  • Another participant expresses appreciation for the references provided, indicating a desire for more comprehensive answers.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the completeness of the explanations provided, and multiple viewpoints regarding the reasons for the chosen representation remain present.

Contextual Notes

Some assumptions about the nature of the CMB power spectrum and its mathematical representation are not fully explored, leaving potential gaps in understanding the implications of the chosen vertical axis.

DoobleD
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Why is the vertical axis in the CMB power spectrum usually chosen as ##l(l+1)C_l/2\pi## instead of simply ##C_l## ?

AHOha.png


The only answer I found come from this post on stackexchange, but the answer doesn't seem very complete. Anyone knows ?
 

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DoobleD said:
Why is the vertical axis in the CMB power spectrum usually chosen as ##l(l+1)C_l/2\pi## instead of simply ##C_l## ?

The only answer I found come from this post on stackexchange, but the answer doesn't seem very complete. Anyone knows ?

Rennie's first reason often is given as the reason, e.g., page 243 from the book "Modern Cosmology" by Scott Dodelson, and section 4.1 from the article "COSMIC BACKGROUND RADIATION MINI-REVIEW" by Scott and Smoot,

https://arxiv.org/abs/astro-ph/0406567

Rennie's second reason also is plausible.
 
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DoobleD said:
Why is the vertical axis in the CMB power spectrum usually chosen as ##l(l+1)C_l/2\pi## instead of simply ##C_l## ?

View attachment 224824

The only answer I found come from this post on stackexchange, but the answer doesn't seem very complete. Anyone knows ?
If the CMB power spectrum were simply a scale-invariant Gaussian random field, then its power spectrum plotted with that vertical axis would be essentially flat (up to variation due to cosmic variance/instrumental noise). I believe the factor ##\ell(\ell+1)## stems from the fact that the spherical harmonics are not strictly flat plane waves.
 
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Awesome, thanks for thoses answers, especially for the references.
 

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