Undergrad Vertical axis of the CMB power spectrum

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SUMMARY

The vertical axis of the Cosmic Microwave Background (CMB) power spectrum is typically represented as ##l(l+1)C_l/2\pi## rather than simply ##C_l## to account for the non-flat nature of spherical harmonics. This representation helps in normalizing the power spectrum, making it easier to interpret the data, especially in the context of cosmic variance and instrumental noise. Key references include "Modern Cosmology" by Scott Dodelson and the article "COSMIC BACKGROUND RADIATION MINI-REVIEW" by Scott and Smoot, which provide foundational insights into this choice.

PREREQUISITES
  • Understanding of CMB power spectrum analysis
  • Familiarity with spherical harmonics
  • Knowledge of Gaussian random fields in cosmology
  • Basic grasp of cosmic variance and instrumental noise
NEXT STEPS
  • Study the mathematical derivation of the CMB power spectrum using spherical harmonics
  • Explore the implications of cosmic variance on CMB measurements
  • Read "Modern Cosmology" by Scott Dodelson for in-depth theoretical context
  • Investigate the role of instrumental noise in astrophysical data analysis
USEFUL FOR

Astronomers, cosmologists, and researchers involved in analyzing CMB data and interpreting power spectrum results.

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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