- #1

fab13

- 318

- 6

- 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

##

\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