Angular power spectrum dependence in redshift z and k

[fab13]

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/

 

[kimbyd]

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

 

[fab13]

@kimbyd .Thanks for your quick answer.

I have looked at your paper and notice that passing :

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.

 

[kimbyd]

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.

 

[fab13]

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

 

[kimbyd]

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.