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I think I can sort of answer your first/second question. It's a bit hard to guess what your background is, but I hope you have seen or derived somewhere that the $a_{lm}$ coefficients can be written as $$\oint \Theta(\hat{x}) Y_{lm}^*(\hat{x}) d\hat{x}$$ where $\Theta$ is the temperature fluctuation (as seen in the CMB) and $Y_{lm}^*$ is (the complex ...

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