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 Y_l^m \sim e^{i m \phi}?

 

[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).