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fab13

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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 :

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

##

\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 :

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

Last edited: