Analytical expression of Cosmic Variance - Poisson distribution?

In summary, the conversation discusses an expression for the Matter Angular power spectrum which includes the spectroscopic bias and the Cosmic variance. The terms in the expression can be computed numerically using a simple rectangular integration method. The speaker has a code that computes the terms for each multipole but is unsure how to compute the Cosmic Variance term. They mention a possible relation for the Signal Noise Ratio and ask for help in finding an expression for the Cosmic Variance to use in the overall calculation. A suggestion is made to check Wikipedia for Sample Variance and to read a paper on uncertainty due to restricted sample size.
  • #1
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TL;DR Summary
I would like to get an analytical expression of Cosmic Variance to be able to compute the Matter Angular Power Spectrum (including theorical signal and spectroscopic Shot Noise). I suspect this Cosmic variane to look like a Poisson distribution but I can't conclude with this up to now.
I have an expression of Matter Angular power spectrum which can be computed numerically by a simple rectangular integration method (see below). I make appear in this expression the spectroscopic bias ##b_{s p}^{2}## and the Cosmic variance ##N^{C}##.

##
\begin{aligned}
\mathcal{D}_{\mathrm{gal}, \mathrm{sp}} &=\left[\int_{l_{m i n}}^{l_{\max }} C_{\ell, \mathrm{gal}, \mathrm{sp}}(\ell) \mathrm{d} \ell\right]=b_{s p}^{2}\left[\int_{l_{\min }}^{l_{\max }} C_{\ell, \mathrm{DM}} \mathrm{d} \ell+N^{C}\right]=b_{s p}^{2}\left[\mathcal{D}_{\mathrm{DM}}+N^{C}\right] \\ & \simeq \Delta \ell \sum_{i=1}^{n} C_{\ell, \mathrm{gal}, \mathrm{sp}}\left(\ell_{i}\right)
\end{aligned}
##

I have a code that computes the terms ##C_{\ell, \mathrm{gal}, \mathrm{sp}}\left(\ell_{i}\right)## for each multipole ##\ell_{i}##. But how to compute the term ##N^{C}##, that is to say, the Cosmic Variance ##N^{C}## :

The only documentation I have found is the following slide from Nico Hamaus :

Capture d’écran 2021-06-13 à 11.52.16.png


But as you can see, I have no explicit expression for Cosmic Variance : Could I consider the relation ##\dfrac{\sigma_{p}}{P}=\dfrac{2}{N_{k}^{1/2}}## as a SNR (Signal Noise Ratio) ?

Which expression of Cosmic Variance could I use to compute the whole expression ##\mathcal{D}_{\mathrm{gal}, \mathrm{sp}}## ?

Thanks in advance, Best regards
 
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  • #2

Related to Analytical expression of Cosmic Variance - Poisson distribution?

1. What is cosmic variance?

Cosmic variance refers to the inherent statistical fluctuations in the distribution of matter and energy in the universe. It is caused by the random nature of the processes that shape the universe, such as the formation of galaxies and the expansion of space.

2. What is an analytical expression?

An analytical expression is a mathematical formula that describes a relationship between variables. It can be used to calculate specific values or make predictions based on the input of certain parameters.

3. How is the Poisson distribution related to cosmic variance?

The Poisson distribution is a mathematical model that describes the probability of rare events occurring within a given time or space. It is often used to analyze cosmic variance because it can accurately predict the frequency of rare cosmic events, such as the formation of galaxy clusters.

4. Why is understanding cosmic variance important?

Understanding cosmic variance is crucial for accurately interpreting and analyzing data from astronomical observations. It allows scientists to distinguish between true variations in the universe and random fluctuations, which can lead to more accurate and reliable conclusions about the nature of the universe.

5. How can the analytical expression of cosmic variance be used in research?

The analytical expression of cosmic variance can be used to calculate the expected level of variance in a given area of the universe, based on the Poisson distribution. This can help researchers determine the significance of their observations and make comparisons between different regions of the universe.

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