A Angular power spectrum dependence in redshift z and k

fab13
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Hi,

I wanted to have a precision about a question that has been post on this relation between P(k) and C_l

The author writes the ##C_\ell## like this :

$$C_\ell(z,z') = \int_0^\infty dkk^2 j_\ell(kz)j_\ell(kz')P(k)$$

I don't undertstand the meaning of ##z## and ##z'## : these are not redshift, are they ?

Normally, it should depend of the multipole ##\ell## but how to make the link between these 2 quantities ##z## and ##z'## and multipole ##\ell##.
Moreover, does the ##P(k)## represent systematically the linear matter power spectrum ? or can we add RSD (Redshift Space Distorsions) like Kaiser or alcock-paczynski effects ?

Thanks for your clarifications and explanations.

Regards
Source https://www.physicsforums.com/threads/relationship-between-the-angular-and-3d-power-spectra.993041/
 
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I think there's a mistake there. I suspect the left hand side is not ##C_\ell##, but the angular correlation function, which is a function of the difference between two angles on the sky, with ##z## and ##z'## being unit vectors.

That said, I don't think this is the way things are currently done. Current power spectrum calculations are based on this paper: https://arxiv.org/abs/astro-ph/9603033
 
@kimbyd .Thanks for your quick answer.

I have looked at your paper and notice that passing :
Capture d’écran 2021-04-14 à 22.12.04.png


From what you seem to say, the ##C(\theta)## on the left would represent the angular correlation function, wouldn't it ?

and in the right term member would appear ##C_{\ell}## which would be the Angular Power spectrum ?

If this is the case, How could I express ##C_{\ell}## from ##C(\theta)##.

Your help will be precious, thanks in advance.
 
Equation 10 there is a reversible transformation. The orthogonality condition is described here:
https://en.wikipedia.org/wiki/Legendre_polynomials#Orthogonality_and_completeness

There's a notational difference there, in part driven by the argument being ##x## rather than ##cos(\theta)##, but it shouldn't be too difficult to take a simple example (e.g. ##P_1##), do the integrals manually, and see what the normalization factor needs to be.
 
Just a precision :

in the link that I gave firstly in my post ( https://www.physicsforums.com/threads/relationship-between-the-angular-and-3d-power-spectra.993041 )

Why does the writter say "How to write the 3D power spectrum, P(k), as an integral of the angular power spectrum, C_l ?"

whereas from kimbyd, it is not the direct relation beween angular power spectrum ##C_{\ell}## and matter power spectrum ##P_{k}##

By, the way, could anyone write the complete formula that links these 2 quantities (##C_{\ell}## and ##P_{k}##) ?

I would be grateful since I am a little confused between angualr correlation function and matter power spectrum and spherical Bessel functions.

Any help would be great, I am begin to desperate.

Best regards
 
I don't think ##P(k)## there is the matter power spectrum. It's still the temperature power spectrum, just represented differently.

The relationship between ##P(k)## and ##C_\ell## is that ##C_\ell## results from the intersection of the 3D power spectrum ##P(k)## with the surface of last scattering.

I believe the equation was written above.
 
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