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fab13

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- 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 :

##\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.

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

##\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.

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