A Computing a variance in astrophysics context

AI Thread Summary
The discussion focuses on computing the variance in the context of astrophysics, specifically regarding photometric galaxy clustering and the covariance of angular power spectra. The introduced observable "O" is defined as the ratio of power matter to angular power spectra, leading to a complex expression that incorporates cosmic variance and shot noise. A key challenge is determining how to compute the variance σ_o² from the simplified expression of "O," particularly due to the square root in its formulation. The conversation highlights confusion around expectations and how to simplify the variance equation while addressing the contributions from different noise terms. Ultimately, the goal is to derive a clear expression for σ_o² based on the provided equations.
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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.
 
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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}## ?
 
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