CMB: L to Angle Conversion Simplified

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In summary, it is often stated that a spherical harmonic l is approximately equal to l = \frac{\pi}{\theta} in discussions about the CMB. This can be demonstrated by choosing a specific Y^l_m with a simple functional form, such as m = \pm l, where the variations are primarily in the \phi direction with a width of \theta = \frac{\pi}{l}.
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In discussion of the CMB it is often claimed that a spherical harmonic l roughly corresponds to [itex]l = \frac{\pi}{\theta}[/itex]. Does anyone know a simple way to show this?
 
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nicksauce said:
In discussion of the CMB it is often claimed that a spherical harmonic l roughly corresponds to [itex]l = \frac{\pi}{\theta}[/itex]. Does anyone know a simple way to show this?
First of all, if we pick any of the various [itex]Y^l_m[/itex]'s, we know that the size of the variations for any [itex]m[/itex] for a given [itex]l[/itex] is the same. So we can pick one particular [itex]Y^l_m[/itex] that has a particularly simple functional form, [itex]m = \pm l[/itex]:

[tex]Y^l_{\pm l} \left(\theta, \phi\right) \propto e^{\pm il\phi}[/tex]

So here we have a situation where all of the variation is in the [itex]\phi[/itex] direction, with the typical with of a peak being [itex]\theta = \frac{\pi}{\l}[/itex].
 

FAQ: CMB: L to Angle Conversion Simplified

What is CMB?

CMB stands for Cosmic Microwave Background. It is the leftover radiation from the Big Bang, and is the oldest light in the universe. It permeates the entire universe and can be observed in all directions.

What is the importance of CMB?

CMB holds important clues about the early universe and its evolution. By studying the characteristics of CMB, scientists can learn more about the age, composition, and structure of the universe.

What is L to Angle Conversion Simplified?

L to Angle Conversion Simplified is a mathematical formula used to convert the angular scale of CMB measurements to physical distances. It takes into account the distance between Earth and the CMB, as well as the angular size of the CMB on the sky.

How is L to Angle Conversion Simplified calculated?

L to Angle Conversion Simplified is calculated by dividing the physical distance between Earth and the CMB by the angular size of the CMB on the sky. This gives the conversion factor between physical distance and angular size, which can then be used to convert CMB measurements to physical distances.

Why is L to Angle Conversion Simplified important?

L to Angle Conversion Simplified is important because it allows scientists to accurately measure the physical size and distance of structures in the early universe. This helps us to better understand the evolution of the universe and the physics that govern it.

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