L to angle conversion

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nicksauce
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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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Chalnoth
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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?
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].
 

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