Relation between a_{\ell m} noise and Poisson noise with C_{\ell}

[fab13]

TL;DR Summary

I would like to make the link between the noise term that appears into $a_{\ell m}$ and the Poisson noise appearing in the "Shot Noise" appearing in a cosmological survey : is $a_{\ell m}$ noise term following a Normal or Poisson distribution ?

We assume two galaxy population, $\mathrm{A}$ and $\mathrm{B}$; the corresponding maps have the following $a_{\ell m}$ :

$\begin{aligned} &a_{\ell m}^{A}=b_{A} a_{\ell m}^{M}+a_{\ell m}^{p A} \\ &a_{\ell m}^{B}=b_{B} a_{\ell m}^{M}+a_{\ell m}^{p B} \end{aligned}$

Here, $b_{A}$ and $b_{B}$ correspond to the linear galaxy bias for both populations, $a_{\ell m}^{p A}$ and $a_{\ell m}^{p B}$ are the respective Poisson contribution to the maps, and $a_{\ell m}^{M}$ is the underlying (identical for both population) dark matter map/distribution.

1 Standard approach

In the "standard analysis", with $a_{\ell m}$ as observables, we have as data vector simply:

$V=\left(\begin{array}{c} a_{\ell m}^{A} \\ a_{\ell m}^{B} \end{array}\right)$

One would then need to compute the covariance matrix of those observable (in order to do some Fisher later on). Here are the various terms:

$\begin{aligned} \operatorname{Cov}\left(a_{\ell m}^{A}, a_{\ell m}^{A}\right) &=b_{A}^{2} \mathcal{C}_{\ell}^{M}+\mathcal{N}_{\ell}^{A} \\ \operatorname{Cov}\left(a_{\ell m}^{A}, a_{\ell m}^{B}\right) &=b_{A} b_{B} \mathcal{C}_{\ell}^{M} \\ \operatorname{Cov}\left(a_{\ell m}^{B}, a_{\ell m}^{B}\right) &=b_{B}^{2} \mathcal{C}_{\ell}^{M}+\mathcal{N}_{\ell}^{B} \\ & \simeq b_{B}^{2} \mathcal{C}_{\ell}^{M} \end{aligned}$

I just want to make the link between $a_{\ell m}^{p A}$ distribution and Shot noise distribution $\mathcal{N}_{\ell}^{A}$ :

How to perform this ? Can random variable $a_{\ell m}^{p A}$ be considered as a Gaussian distribution and $\mathcal{N}_{\ell}^{A}$ a Poisson distribution ( $\mathcal{N}_{\ell}^{A}=\dfrac{1}{n_{A}^{2}})$ with $n_A$ the density of galaxies.
Any help is welcome

 

[AndyW]

.

Thank you for your question. In order to make the link between $a_{\ell m}^{p A}$ and $\mathcal{N}_{\ell}^{A}$, we first need to define what these terms represent.

$\mathcal{N}_{\ell}^{A}$ is the shot noise contribution to the covariance matrix, which arises from the Poisson distribution of galaxies in the observed map. This means that the number of galaxies in each pixel of the map is a random variable, following a Poisson distribution with mean $n_A$, the average density of galaxies in the population A. Therefore, the shot noise contribution to the covariance matrix is given by $\mathcal{N}_{\ell}^{A}=\dfrac{1}{n_{A}^{2}}$.

On the other hand, $a_{\ell m}^{p A}$ represents the Poisson contribution to the observed map. This term arises from the fact that the observed map is a combination of the underlying dark matter map $a_{\ell m}^{M}$ and the Poisson contribution from the galaxies in population A. In other words, it accounts for the fluctuations in the observed map due to the discrete nature of galaxies.

Therefore, we can see that there is a direct link between $a_{\ell m}^{p A}$ and $\mathcal{N}_{\ell}^{A}$, as they both represent the Poisson contribution to the observed map. However, it is important to note that $a_{\ell m}^{p A}$ is a random variable, while $\mathcal{N}_{\ell}^{A}$ is a fixed value calculated from the average density of galaxies in population A.

I hope this helps to clarify the relationship between these terms. Please let me know if you have any further questions.