# Shot noise in cosmology : Expectation and Variance expression

I am trying to express the Shot noise in cosmology context, mostly the Shot noise expression into error on a $$C_\ell$$

If this would be only for myself, I would take $$\text{Variance}(Shot\,noise) = 1/n_{gal}$$ with $$n_\text{gal}$$ the average density of galaxies.

But my tutor disagreess. Actually, the formula that puts the mess is the following one :

$C_\ell$" />

I am only interested in the term A=B, that is to say $$\Delta_{ij}^{GG}$$.

So we have from eq(138) : $$\Delta_{ij}^{GG} = \sqrt{\dfrac{2}{(2\ell+1)f_{sky}\Delta\ell}}[C_ij^{GG}+N_{ij}^{GG}]$$

I don't understand why the standard deviation for shot noise in eq(138) is defined as $$N_{ij}^{GG}=\dfrac{1}{n_{i}}\delta_{ij}^{K}$$ and not $$N_{ij}^{GG}=\dfrac{1}{\sqrt{n_{i}}}\delta_{ij}^{K}$$.

Such way, I could have the $$\text{variance}(P_{shot}) = \dfrac{1}{n_{gal}}$$.

One has to not forget that $$\Delta_{ij}^{GG}$$ is a standard deviation and not a variance.

Do you understand my issue ?

Any help is welcome

• It's not clear to me what you mean by "Shot Noise", particularly as this is usually a synonym for noise that follows Poisson statistics. (e.g., en.wikipedia.org/wiki/Shot_noise) – Peter Erwin Jun 1 at 12:51
• @PeterErwin the OP notes the connection "since Shot Noise is often assimilated to a Poisson noise" but at least the basic Poisson distribution requires a $(k!)^{-1}$ where $k$ is an integer, and I think the analysis discussed here is of density fluctuations over various scales and so there aren't discreet volumes with integer numbers of galaxies in them, except perhaps within some integral over volume sizes. – uhoh Jun 2 at 4:06
• @uhoh A Poisson distribution has (using the symbols from the Wiki page) an expectation value of $\lambda$ (not $k$) and a variance of $\lambda$. For the "density of galaxies" problem, $\lambda = \bar{n}_{\rm gal}$, so I don't understand why the OP says the "Shot Noise" should be $1/\bar{n}_{\rm gal}$. – Peter Erwin Jun 2 at 9:26
• @PeterErwin in backwards order; yes I see, the variance $\sigma^2$ of a Poisson distribution is equal to $\lambda$ just like the normal distribution. About $\lambda$ and $k$, I'm just thinking of how I've used them; I had data of integer values $k$ and looked for the likelihood that it was the result of some value $\lambda$; finding the most likely $\lambda$ to have produced the measured $k$ (maximum likelihood). – uhoh Jun 2 at 10:18
• @uhoh . Thanks for your answer. Unfortunately, I don't know yet how to justify that the expectation of Shot Noise $\bar{N}_{sp}$ should be equal to 0 like a white noise. I am also interested in the variance of Shot Noise : if I take a fixed value for the spectroscopic Shot Noise, then, the variance is equal to : $<(N_{sp}-\bar{N}_{sp})^{2}>=1/\bar{n}_{\rm gal}^{2}$. Could anyone confirm this result ? Regards – youpilat13 Jun 2 at 10:34