A Poisson noise on ##a_{\ell m}## complex number: real or complex?

AI Thread Summary
In the context of cosmology, the discussion revolves around the nature of Poisson noise added to the complex coefficients ##a_{\ell m}##. It questions whether this noise is considered a complex or real number, with the variance of the Poisson noise defined as ##\text{Var}(\bar{a}_{\ell m}^{p}) = \dfrac{1}{n_{gal}\,f_{sky}}##. Additionally, the variance of the real and imaginary parts of ##a_{\ell m}## is examined, suggesting that each has a variance of ##\dfrac{C_\ell}{2}##, given that ##a_{\ell m}## is complex. The relationship between the magnitudes and variances of these components is also discussed, emphasizing the need for clarification on these points. Overall, the thread seeks to clarify the implications of Poisson noise on the statistical properties of the coefficients in cosmological analyses.
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I try to get clarifications about the Poisson's noise with spherical harmonics of Legendre transformation
1) In a cosmology context, when I add a centered Poisson noise on ##a_{\ell m}## and I take the definition of a ##C_{\ell}## this way :

##C_{\ell}=\dfrac{1}{2\ell+1} \sum_{m=-\ell}^{+\ell} \left(a_{\ell m}+\bar{a}_{\ell m}^{p}\right)\left(a_{\ell m}+\bar{a}_{\ell m}^{p}\right)^* ##

Is Poisson noise a complex number or is it simply a real number ? knowing that variance of Poisson is equal in my case :

##\text{Var}(\bar{a}_{\ell m}^{p}) = \dfrac{1}{n_{gal}\,f_{sky}}## where ##n_{gal}## the density of galaxies and ##f_{sky}## the fraction of sky observed.

I work with fluctuations of matter density (not temperature fluctuations).

2) What is the variance of real part and imaginary part of an ##a_{\ell m}## : usually, one says that :

##\text{Var}(a_{\ell m}) = C_{\ell}## but given the fact that ##a_{\ell m}## is a complex number, we could say that :

##\text{Var}(\text{Re}(a_{\ell m}))## has a variance equal to ##\dfrac{C_\ell}{2}##

and

##\text{Var}(\text{Im}(a_{\ell m}))## has a variance equal to ##\dfrac{C_\ell}{2}##

since :

##\begin{aligned}
& \left|a_{\ell m}\right|^2=\operatorname{Re}\left(a_{\ell m}\right)^2+\operatorname{Im}\left(a_{\ell m}\right)^2 \\
& E\left[\left|a_{\ell m}\right|^2\right]=E\left[\operatorname{Re}\left(a_{\ell m}\right)^2\right]+E\left[\operatorname{Im}\left(a_{\ell m}\right)^2\right]=C_{\ell}
\end{aligned}##

Is it correct ?

Any clarification is welcome.
 
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