Photometric Galaxy Clustering Error and Poisson Noise

[fab13]

TL;DR Summary

I would like to make the connection between the expression of squared for error on photometric galaxy clustering and the definition of variance of Shot noise. The issue lies in the expression of covariance between $C_\ell$ and $B_{sp}$ which may be equal to zero but I am not at 100% since I don't know how to justify that the product $C_\ell\,B_{sp}$ is null. Any clarification is welcome

The error on photometric galaxy clustering under the form of covariance which is actually a standard deviation expression for a fixed multipole $\ell$ :

$\sigma_{C, i j}^{A B}(\ell)=\Delta C_{i j}^{A B}(\ell)=\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}}\left[C_{i j}^{A B}(\ell)+N_{i j}^{A B}(\ell)\right]\quad(1)$

where $f_{\text {sky }}\simeq 0.36$ is the fraction of survey sky and $A, B$ run over the observables $L$ and $G$ : We look at here $A, B=G$.

We recall that $\Delta \ell$ is the width of the multipoles bins used when computing the angular power spectra, and $i, j$ run over all tomographic bins.

The First $\operatorname{term} C_{i j}^{A B}$ refers to the raw angular power spectrum including what we call the "cosmic variance" : this is an intrinsic noise due to the fact that Universe has a finite volume.

The second $\operatorname{term} N_{i j}^{A B}(\ell)$ is the residual Shot Noise (Poisson noise) :

1. I make confusion between $N_{i j}^{A B}(\ell)$ and Shot noise $B_{s p}$ and $B_{p h}$ for which we add a prefactor $\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}} .$

Can I write from $(1)$ :

$\begin{aligned} &\operatorname{Var}\left(B_{s p, i}(\ell)\right)=\left\langle B_{s p, i}(\ell)^{2}\right\rangle=\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell} N_{s p}^{2}(\ell) \quad \text{ with}\quad N_{s p}(\ell)=\frac{1}{\bar{n}_{s p, i}} \\ &\operatorname{Var}\left(B_{p h, i}(\ell)\right)=\left\langle B_{p h, i}(\ell)^{2}\right\rangle=\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell} N_{p h}^{2}(\ell) \quad \text{ with} \quad N_{p h}(\ell)=\frac{1}{\bar{n}_{p h, i}} \end{aligned}\quad(2)$

??

Indeed, taking the square of $(1)$ for the case $N_{ij}^{GG}\equiv N_{sp}$ make appear cross-terms :

$Var(C_\ell)=(\sigma_{C, i j}^{GG}(\ell))^{2}=\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell} \bigg[C_\ell^2+2C_\ell\,N_{sp}+N_{s p}^{2}\bigg]$ and I don't know how to make the connection with $\text{Var}(B_{sp})$.

Maybe I could make the link with the expression :

$\operatorname{Var}[\mathrm{X}+\mathrm{Y}]=\operatorname{Var}[\mathrm{X}]+\operatorname{Var}[\mathrm{Y}]+2 \cdot \operatorname{Cov}[\mathrm{X}, \mathrm{Y}]$

and do the link with :

$\text{Var}(C_\ell+N_{sp}) = \text{Var}(C_\ell) + 2 \text{Cov}(C_\ell,N_{sp}) + \text{Var}(N_{sp})$

But how to prove that : $2\,\text{Cov}(C_\ell,N_{sp})=2\,C_\ell\,N_{sp}$ ??

If I could prove it, this way, Maybe we could infer the expression of variance for integrated Shot noise over $\ell$ (noted $B_{s p, i n t}$ and $B_{p h, i n t}$) and with subscript $i$ referencing the $i$-th bin of redshift :
$\begin{gathered} \operatorname{Var}\left(B_{s p, i n t, i}\right)=\operatorname{Var}\left(\sum_{j=1}^{N} \Delta \ell B_{s p, i}\left(\ell_{j}\right)\right)=\Delta \ell^{2} \sum_{j=1}^{N} \operatorname{Var}\left(B_{s p, i}\right) \\ =\sum_{\ell=\ell_{m i n}}^{\ell_{\max }} \frac{2 \Delta \ell}{(2 \ell+1) f_{\mathrm{sky}}} \frac{1}{\bar{n}_{s p, i}^{2}} \\ \operatorname{Var}\left(B_{p h, i n t, i}\right)=\operatorname{Var}\left(\sum_{j=1}^{N} \Delta \ell B_{p h, i}\left(\ell_{j}\right)\right)=\Delta \ell^{2} \sum_{j=1}^{N} \operatorname{Var}\left(B_{p h, i}\right) \\ =\sum_{\ell=\ell_{m i n}}^{\ell_{m a x}} \frac{2 \Delta \ell}{(2 \ell+1) f_{\mathrm{sky}}} \frac{1}{\bar{n}_{p h, i}^{2}} \end{gathered}\quad(3)$

2. But after these 2 expression of variance in $(3)$, I am disturbed in my computation by the factor $\Delta\ell$ in the numerator, that is to say, my results are not consistent.

An ideal case for the integration of $B_{sp,i}$ and $B_{ph,i}$ would be to take $\Delta\ell=1$ but I don't know if it is correct knowing that I am using a larger value $\Delta\ell$ for angular power spectrum (actually, $\Delta\ell=80$ in my code since we have a value for $C_\ell$ every 80 multipoles $\ell$). So I don't know if I am coherent.

Sorry if have been a little bit long but this was necessary for people interested by this post.

Any help is welcome.

 

[AndyW]

Hello,

Thank you for your detailed forum post. I am a scientist and I would be happy to help clarify some of the confusion you are experiencing.

Firstly, I would like to address your question about the connection between $\operatorname{Var}(B_{sp})$ and the expression you have provided in $(1)$. You are correct in your understanding that the expression in $(1)$ is for the covariance of the angular power spectrum, and not the variance of the shot noise. To calculate the variance of the shot noise, you would need to take the square of the shot noise term in $(1)$, which would result in the expression you have provided in $(2)$.

Now, moving on to your question about the factor of $\Delta\ell$ in the numerator in your expression for the integrated shot noise in $(3)$. This factor is necessary because you are integrating over a range of multipoles, and therefore you need to account for the fact that the shot noise will vary with the width of the multipole bins. In other words, the shot noise will be different for different $\Delta\ell$ values, and the factor of $\Delta\ell$ ensures that you are accounting for this variation.

I understand your concern about using a larger value for $\Delta\ell$ in your code, but this is necessary for accurate calculations. In fact, the larger the value of $\Delta\ell$, the more accurate your results will be. This is because a larger $\Delta\ell$ means that you are integrating over a wider range of multipoles, which will give you a more accurate representation of the true shot noise.

I hope this helps to clarify your confusion. Please let me know if you have any further questions.