# Spherical harmonics - Computing the variance of Poisson noise integrated over $\ell$ on a defined quantity?

It is an astrophysics context but actually, it is mostly a mathematics issue.

From spherical harmonics with Legendre deccomposition, I have the following definition of the standard deviation of a $$C_\ell$$ noised with a Poisson Noise $$N_p$$ :

$$$$\sigma({C_\ell})(\ell)=\sqrt{\frac{2}{(2 \ell+1) f_{sky}}}\left[C_\ell(\ell)+\dfrac{1}{N_{p}}\right]\label{1}\tag{1}$$$$

Now I consider the quantity : $$\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}$$

I want to estimate the variance expression of Poisson Noise of this qantity.

For that, I take the definition of $$a_{lm}$$ following a normal distribution with mean equal to zero and take also the definition of a $$C_\ell=\langle a_{lm}^2 \rangle=\dfrac{1}{2\ell+1}\sum_{m=-\ell}^{\ell}\,a_{\ell m}^2 = \text{Var}(a_{lm})$$.

I use $$\stackrel{d}{=}$$ to denote equality in distribution : $$\begin{split} Z & \equiv \sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{m=-\ell}^{\ell} a_{\ell, m}^{2} \\ & =\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{m=-\ell}^{\ell} C_{\ell} \cdot\left(\frac{a_{\ell, m}}{\sqrt{C_{\ell}}}\right)^{2} \\ & \sim \sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{m=-\ell}^{\ell} C_{\ell} \cdot \mathrm{ChiSq}(1) \\ &=\sum_{\ell=\ell_{\min }}^{\ell_{\max }} C_{\ell} \cdot \mathrm{ChiSq}(2 \ell+1). \end{split}$$ So Finally we have : $$\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2} \stackrel{d}{=} \sum_{\ell=1}^{N}\,C_\ell \chi^{2}(2\ell+1).$$ To compute the Poisson Noise of each $$a_{\ell m}^2$$, a colleague suggests me that for getting an optimal variance expression, I must use the Inverse-variance_weighting, to take only the quantity below: $$\dfrac{2}{f_{sky}\,N_p^2},\label{2}\tag{2}$$ and to do the summation to get the variance of Poisson Noise (to make the link with formula \eqref{1} and be consistent with it) : $$\text{Var}(N_{p,int})=\sum_{\ell=1}^{N} \dfrac{2}{f_{sky}\,N_p^2}. \label{3}\tag{3}$$ But I have difficulties to understand the Inverse-variance_weighting applied in my case : I don't understand where are implied the weights regarding the formula \eqref{3}.

I wrote above $$\text{Var}(N_{p,int})$$, make caution, this is just to express the "integrated Poisson variance" over $$\ell$$ in the quantity $$\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}.$$ Unfortunately, when I do numerical computation with this variance formula \eqref{3}, I don't get the same variance than with another valid method.

Correct results are got by introducing a factor $$\sqrt{2\ell+1}$$ into formula \eqref{3}. But I don't know how to justify it, it is just fine-tuned from my part for the moment and it is not rigorous I admit.

QUESTION :

1. Is formula \eqref{2} that expresses the variance of Shot Noise on the quantity $$\sum_{\ell=\ell_{min}}^{\ell_{max}} \sum_{m=-\ell}^\ell a_{\ell,m}^2$$ correct ?

If yes, how could integrate it correctly by summing it on multipole $$\ell$$ and to remain consistent with the standard deviation formula \eqref{1} ? (I talk about the pre-factor $$\sqrt{\frac{2}{(2 \ell+1) f_{sky}}}$$ that disturbs me in the expression of integrated Poisson Noise)

2. Finally, from all given informations above, how to correctly compute analytically the variance of

$$\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}\:?$$