Fourier transform and Cosmic variance - a few precisions

[fab13]

TL;DR Summary

I would like to understand the reasoning which is done on a report about the cosmic variance. I nedd precisions to know how the expression linking the error relative of Power spectrum and the number of pixels in Fourier space. I would like also to understand under which conditions relative error and standard deviation are equal.

I cite an original report of a colleague :

If we are interested in power spectrum, we want to estimate the
variance of the amplitude of the modes $k$ of our Fourier
decomposition. If one observes the whole observable Universe and we
do the Fourier transformation we get a cube whose center is the mode
$|\vec{k}| = 0$ which corresponds to the mean value of the observed
field.

This mode has only one pixel. How do we measure the variance of the
process at $|\vec{k}| = 0$ ? We can not.

In fact we can but the value of doesn't mean anything because the
error is infinite. In other words, we have an intrinsic (statistical)
error which depends on the number of achievements to which we have
access.

one will be able to consider spheres of sizes $dk$ between $[k: k + dk]$ which will contain a number of pixels $N_ {k} = V_ {k}/(dk)^{3}$ where $V_{k} = 4 \pi k^{2} dk$ is the volume of the sphere
and $(dk)^{3}$ is the volume of a pixel in our Fourier transform
cube.

So one can estimate how many values one can use to calculate our
power spectrum for each value of $k$. The greater the $k$, the
greater the number of accessible values and therefore the statistical
error decreases. The power spectrum is a variance estimator so the
statistical error is basically a relative error:

$\dfrac{\sigma (P(k))}{P(k)} = \sqrt{\dfrac{2}{N_{k} -1}}_{\text{with}} N_{k} \approx 4\pi \left (\dfrac{k}{dk}\right)^{2}$

So we can see that for the case $|\vec{k}| = 0$ we have an infinite error because $N_{k} = 1$.

1) I can't manage to proove that the statistical error is formulated like :

$\dfrac{\sigma (P (k))}{P(k)} = \sqrt{\dfrac {2}{N_{k} -1}}_{\text{with}} N_{k} \approx 4\pi \left(\dfrac{k}{dk}\right)^{2}$

and why it is considered like a relative error ?

2) Which are the conditions to assimilate a statistical error (standard deviation) to a relative error ($\dfrac{\Delta x}{x}$) ?

3) How to prove that $N_{k} = 1$ in the case $|\vec{k}| = 0$ ?

Any help is welcome.

 

[PeterDonis]

I cite an original report of a colleague

Please give a reference. You can't cite a source that nobody else but you can see.

 

[fab13]

Sorry, I give you the link of initial discussion about my question but you won't surely understand the text part since it is in french :

https://forums.futura-sciences.com/...gnification-variance-cosmique-shot-noise.html

Hoping this will help you

 

[fab13]

Isn't really no one who could help me about this problem of understanding ?

 

[PeterDonis]

Hoping this will help you

Aside from the text being in French, this is still just a discussion forum and does not give enough information to figure out what the discussion is supposed to be about. There is one link to what looks like it should be a presentation, but the link is not valid (404 error).

Is there a peer-reviewed paper or something similar that the discussion you linked to is based on? Can you provide a link to such a paper?