I never able to wrap my head around alm term used in expression enter image description here

If you know the expression, kindly explain but also give textbook references. It would be helpful.


1 Answer 1


Any function on a sphere can be expanded as a linear combination of spherical harmonics $Y_{\ell m}$. The numbers $a_{\ell m}$ tell you “how much” of each spherical harmonic is needed. This is similar to a Fourier series for decomposing any periodic function $f(x)$ into its sinusoidal “harmonics”.

Conceptually, the spherical harmonics are a complete orthonormal basis for the infinite-dimensional vector space of functions on a sphere, in the same way that $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are a complete orthonormal basis for three-dimensional Euclidean vector space (and in the same way that $\cos{nx}$ and $\sin{nx}$ are a complete orthonormal basis for periodic functions in one dimension). The $a_{\ell m}$ are the components of $T$ in this basis.

One classic reference book for such things is Methods of Mathematical Physics by Courant and Hilbert.

The book or paper you referenced has a typo. They meant that the $a_{00}$ term measures the mean temperature. This is because the first spherical harmonic $Y_{00}$ is a constant over the sphere. The other terms represent angular fluctuations in temperature around the mean, on smaller and smaller angular scales as $\ell$ increases.

  • $\begingroup$ nice answer. +1. Of course I knew exactly that, but I couldn't have said it so clearly or succinctly. $\endgroup$ Jul 31, 2020 at 16:54
  • $\begingroup$ Good answer sir, i got your point but i am asking for the mathematics behind it. And also textbook references for the expression. $\endgroup$ Jul 31, 2020 at 19:22
  • $\begingroup$ @RajdeepSingh The mathematics is summarized in the first and second paragraphs, and a textbook reference is provided in the third. The relevant concepts to learn about are abstract vector spaces (including spaces of functions as infinite-dimensional examples), bases, orthonormality, and completeness. All of these are standard concepts in, say, a course on quantum mechanics. Have you studied QM yet? All these concepts are also involved in the classical mechanics of, say, a vibrating string. Did you learn about Fourier analysis then? $\endgroup$
    – G. Smith
    Jul 31, 2020 at 19:49
  • $\begingroup$ @RajdeepSingh textbook references for the expression An example of your expression is in Wikipedia (and surely also in Courant and Hilbert’s book). This part of the article also explains how to use the orthonormality of the spherical harmonics to calculate the $a_{\ell m}$ from the function $T$. This expansion has essentially nothing to do with temperature or the CMB, in case this was something you were wondering about. Any function on a sphere can be expanded like this. $\endgroup$
    – G. Smith
    Jul 31, 2020 at 20:04
  • $\begingroup$ @RajdeepSingh If you want to understand why the spherical harmonics are a complete orthonormal basis for functions on a sphere, and how they were discovered, then the relevant math is Sturm-Liouville theory as applied to Laplace’s equation on a sphere. Courant and Hilbert probably covers this in detail. I’m sorry that I can’t say for sure; I used to have a copy but no longer do. $\endgroup$
    – G. Smith
    Jul 31, 2020 at 20:17

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