CMB , Spherical Harmonics and Rotational Invariance

[center o bass]

In Dodelson's "Modern Cosmology" on p.241 he states that the ##a_{lm}##-s -- for a given ##l##-- corresponding to a spherical harmonic expansion of the photon-temperature fluctuations, are drawn from the same probability distribution regardless of the value of ##m##. Dodelson does not explain this any further, but other authors claim that it is due to the fact that ##m## somehow corresponds to an orientation and this should not matter as the universe is (believed to be) statistically rotational invariant.

Question:
What is the precise property of the spherical harmonic ##Y_l^m## for a given ##l## that justifies this claim?

 

[ChrisVer]

That [itex]Y_l^m \sim e^{i m \phi}[/itex]?

 

[Chalnoth]

Question:
What is the precise property of the spherical harmonic ##Y_l^m## for a given ##l## that justifies this claim?

The ##Y_\ell^m## functions for a given ##\ell## can be morphed into one another through rotations in any direction. That is, if you rotate the coordinate system, the resulting ##a_{\ell m}## parameters are a linear combination of the pre-rotated ##a_{\ell m}## parameters. During this rotation, only the ##a_{\ell m}## values with the same ##\ell## are mixed.

 

[center o bass]

The ##Y_\ell^m## functions for a given ##\ell## can be morphed into one another through rotations in any direction. That is, if you rotate the coordinate system, the resulting ##a_{\ell m}## parameters are a linear combination of the pre-rotated ##a_{\ell m}## parameters. During this rotation, only the ##a_{\ell m}## values with the same ##\ell## are mixed.

Thanks for the reply! From what you've now said, how would one go on to argue (fairly rigorously) that the ##a_{lm}##-s for a given ##l## must be drawn from the same probability distributions?

 

[Chalnoth]

Thanks for the reply! From what you've now said, how would one go on to argue (fairly rigorously) that the ##a_{lm}##-s for a given ##l## must be drawn from the same probability distributions?

That's the assumption of isotropy. As the different coefficients for the same ##\ell## are just rotations of one another, assuming isotropy requires that they all have the same probability distribution (provided you make use of the appropriate normalization for the ##Y_l^m## functions).