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 $N_p$ :
\begin{equation} \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} \end{equation}
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 :
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)Finally, from all given informations above, how to correctly compute analytically the variance of
$$\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}\:?$$
