Computing a variance in astrophysics context

[fab13]

TL;DR

Computing a variance in astrophysics context : the goal is to compute the variance of a ratio of 2 parameters into astrophysics context. I have posted here since the issue is about statistics.

Below the error on photometric galaxy clustering under the form of covariance :

$$

\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]

$$

where $_{\text {sky }}$ is the fraction of surveyed sky and $A, B$ run over the observables ##$and$, \Delta \ell$is the width of the multipoles bins used when computing the angular power spectra, and$, j$run over all tomographic bins. The First term$_{i j}^{A B}$refers to the Cosmic Variance and the second term$_{i j}^{A B}(\ell)$is the Shot Noise (Poisson noise). We look at here$, B=G$. We introduce a new observable called "O"which is the ratio between power matter and angular power spectra O=\left(\frac{C_{\ell, \mathrm{gal}, \mathrm{sp}}^{\prime}}{C_{\ell, \mathrm{gal}, \mathrm{ph}}^{\prime}}\right)^{1 / 2}=\left(\frac{b_{s p}}{b_{p h}}\right) Taking the ratio between both, one can write : O=\left(\frac{b_{s p}^{2} C_{\ell, \mathrm{DM}}^{\prime}+\Delta C_{s p, i j}^{G G}}{b_{p h}^{2} C_{\ell, \mathrm{DM}}^{\prime}+\Delta C_{p h, i j}^{G G}}\right)^{1 / 2} We neglect the Poisson noise term$\Delta C_{p h, i j}^{G G}$(sum of Cosmic Variance and Shot Noise)$\Delta C_{p h, i j}^{G G}$on denominator since it is very small compared to$b_{p h}^{2} C_{\ell, \mathrm{DM}}^{\prime}$We consider also the dominance of spectroscopic Shot Noise$N_{s p, i i}^{G G}(\ell)$in the quantity$\Delta C_{s p, i j}^{G G}$Let's recall the notation for photometric$C_{\ell, \text { gal }, \mathrm{ph}}^{\prime}$: C_{\ell, \mathrm{gal}, \mathrm{ph}}^{\prime}=\int_{l_{m i n}}^{l_{\max }} C_{\ell, \mathrm{gal}, \mathrm{ph}}(\ell) \mathrm{d} \ell=b_{p h}^{2} \int_{l_{\min }}^{l_{\max }} C_{\ell, \mathrm{DM}} \mathrm{d} \ell=b_{p h}^{2} C_{\ell, \mathrm{DM}}^{\prime} This way, one has : O=\left(\frac{b_{s p}^{2} C_{\ell, \mathrm{DM}}^{\prime}+N_{s p, i j}^{G G}(\ell)}{b_{p h}^{2} C_{\ell, \mathrm{DM}}^{\prime}+\Delta C_{p h, i j}}\right)^{1 / 2}=\left(\frac{b_{s p}^{2}}{b_{p h}^{2}}+\frac{\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}} N_{s p, i j}^{G G}(\ell)}{C_{\ell, \mathrm{gal}, \mathrm{ph}}^{\prime}(\ell)}\right)^{1 / 2} and finally for each bin$$: \sigma_{o}^{2}=\left[\int_{l_{m i n}}^{l_{\max }} C_{\ell, \mathrm{gal}, \mathrm{ph}}(\ell) \mathrm{d} \ell\right]^{-1}\left[\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}\right]^{1 / 4}\left(N_{s p, i j}^{G G}(\ell)\right)^{1 / 2} with : N_{s p, i j}^{\mathrm{GG}}(\ell)=\frac{1}{\bar{n}_{s p, i}} \delta_{i j}^{\mathrm{K}} with$\bar{n}_{s p, i}$the spectroscopic density of galaxies per bin.QUESTION: How to compute the variance$\sigma_o^2## from the last simplified expression of the ratio.The issue comes from the fact that I have a square root in my expression for the observable "0" :\begin{equation}

O=\left(\frac{C_{\ell, \mathrm{gal}, \mathrm{sp}}^{\prime}}{C_{\ell, \mathrm{gal}, \mathrm{ph}}^{\prime}}\right)^{1 / 2}=\left(\frac{b_{s p}}{b_{p h}}\right)

\end{equation}I have posted on https://math.stackexchange.com/questions/4087630/variance-of-a-the-root-square-of-a-quantity but from the answer :\begin{align}\operatorname{Var}X&=\Bbb E(X^2)-(\Bbb EX)^2\\&=\Bbb E(b_1^2/b_2^2+N/f)-\left(\Bbb E\sqrt{b_1^2/b_2^2+N/f}\right)^2\\&=b_1^2/b_2^2+\Bbb E(N/f)-\left(\Bbb E\sqrt{b_1^2/b_2^2+N/f}\right)^2.\end{align} I have to compute expectation and I don't know how to compute these expectations (relatively to which quantity ? on $\ell$ multipole ? on Observable Covariance $C_{ij}^{AB}$ ?It is confused in my head, if someone could help me or gives suggestions, this would be fine.

 

[fab13]

I have just realized tha the simplifed expression :

$$\sigma_{o}^{2}=\left[\int_{l_{m i n}}^{l_{\max }} C_{\ell, \mathrm{gal}, \mathrm{ph}}(\ell) \mathrm{d} \ell\right]^{-1}\left[\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}\right]^{1 / 4}\left(N_{s p, i j}^{G G}(\ell)\right)^{1 / 2}\quad(1)$$

is wrong, I can't infer this from the previous one :

$$O=\left(\frac{b_{s p}^{2} C_{\ell, \mathrm{DM}}^{\prime}+N_{s p, i j}^{G G}(\ell)}{b_{p h}^{2} C_{\ell, \mathrm{DM}}^{\prime}+\Delta C_{p h, i j}}\right)^{1 / 2}=\left(\frac{b_{s p}^{2}}{b_{p h}^{2}}+\frac{\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}} N_{s p, i j}^{G G}(\ell)}{C_{\ell, \mathrm{gal}, \mathrm{ph}}^{\prime}(\ell)}\right)^{1 / 2}\quad(2)$$

How can I simplify the equation (2) to get the expression of $\sigma_o^{2}$ ?