CMB: L to Angle Conversion Simplified

[nicksauce]

In discussion of the CMB it is often claimed that a spherical harmonic l roughly corresponds to l = \frac{\pi}{\theta}. Does anyone know a simple way to show this?

 

[Chalnoth]

In discussion of the CMB it is often claimed that a spherical harmonic l roughly corresponds to l = \frac{\pi}{\theta}. Does anyone know a simple way to show this?

First of all, if we pick any of the various Y^l_m's, we know that the size of the variations for any m for a given l is the same. So we can pick one particular Y^l_m that has a particularly simple functional form, m = \pm l:

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

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