Computing a variance in astrophysics context

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• fab13
In summary, the conversation discusses the error on photometric galaxy clustering and introduces a new observable called "O" which is the ratio between power matter and angular power spectra. The conversation also talks about how to compute the variance of this new observable, with the simplified expression in equation (1) being incorrect. The correct expression needs to be inferred from the previous equation (2).
fab13
TL;DR Summary
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" :

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)

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.

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}## ?

1. What is the purpose of computing a variance in astrophysics?

The purpose of computing a variance in astrophysics is to measure the spread or variability of a set of data points. This is important in understanding the precision and accuracy of measurements and in comparing different data sets.

2. How is the variance calculated in astrophysics?

In astrophysics, the variance is calculated by taking the average of the squared differences between each data point and the mean of the data set. This value is then used to determine the spread of the data points around the mean.

3. What are the units of variance in astrophysics?

The units of variance in astrophysics are typically expressed as the square of the original units of the data points. For example, if the data points are measured in meters, the variance will be expressed in square meters.

4. How is the variance used in astrophysics research?

The variance is used in astrophysics research to assess the reliability and significance of data sets. It can also be used to identify any outliers or unusual data points that may affect the overall analysis.

5. Can the variance be negative in astrophysics?

No, the variance cannot be negative in astrophysics. It is always a positive value as it represents the squared differences between data points and the mean. A negative value would indicate that the data points are closer to the mean than expected, which is not possible.

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