Consider an isotropic random gaussian $f(\vec{\theta})$ on the surface of a unit sphere. We can decompose $f(\vec{\theta})$ in 2 ways.
- $f(\vec\theta) = \int \frac{d^2 l}{(2\pi)^2} \exp^{i \vec l \cdot \vec \theta} \tilde f(\vec l)$
- $f(\vec\theta) = \sum_{lm} a_{lm}Y_{lm}(\vec \theta)$
One way to define a power spectrum $C_l$ is by looking at the correlation in $\vec l$ space: $<\tilde f(\vec l) \tilde f(\vec l’)> = (2\pi)^2 \delta_{Dirac}(\vec l + \vec l’) C_l$
Another way is to use the coefficients of the spherical harmonics $<a_{lm}a_{l’m’}> = \delta_{ll’}\delta_{mm’}C_l$
($C_l$ in both cases only depends on the magnitude of $l$ because of isotropy)
My question is that the 2 definitions of the power spectra seem to have the same spirit, however, I am having trouble to show their equivalency or how they might be related to each other. Also, is it true that $\tilde f(\vec l)$ is only non-zero when $\vec l$ has integer magnitude? Thank you in advance!