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- Summary:
- I would like to go deeper in the relationship between Matter power spectrum and Angular power spectrum.

From a previous post about the Relationship between the angular and 3D power spectra , I have got a demonstration making the link between the Angular power spectrum ##C_{\ell}## and the 3D Matter power spectrum ##P(k)## :

##

C_{\ell}\left(z, z^{\prime}\right)=\int_{0}^{\infty} d k k^{2} j_{\ell}(k z) j_{\ell}\left(k z^{\prime}\right) P(k)

##

where ##j_{\ell}## are the spherical Bessel functions.

Given

## \tag{1}

C_{\ell}\left(z, z^{\prime}\right)=\int_{0}^{\infty} d k k^{2} j_{\ell}(k z) j_{\ell}\left(k z^{\prime}\right) P(k)

##

Question: how to invert the integral to find the function ##P(k)##?

==>

The closure relation for spherical Bessel function:

## \tag{2}

\int_0^\infty x^2 j_n(xu) j_n(xv) dx = \frac{\pi}{2u^2} \delta(u-v).

##

Multipy Eq.(1) with ##z^2 j_\ell(qz)## and integral over ##z##:

\begin{align}

\int_0^\infty z^2 j_\ell(qz) C_{\ell}\left(z, z^{\prime}\right) dz =&\int_{0}^{\infty} d k k^{2} \left\{ \int^0_\infty z^2 dz j_\ell(qz) j_{\ell}(k z)\right\} j_{\ell}\left(k z^{\prime}\right) P(k) \\

=&\int_{0}^{\infty} d k k^{2} \left\{\frac{\pi}{2q^2} \delta(q-k)\right\} j_{\ell}\left(k z^{\prime}\right) P(k) \\

=& q^{2} \frac{\pi}{2q^2} j_{\ell}\left(q z^{\prime}\right) P(q) \tag{3}.

\end{align}

Once again multiply Eq.(3) with ##z'^2 j_\ell(q'z')## and integral over ##z'##

\begin{align}

\int_0^\infty z'^2 dz' j_\ell(q'z') \int_0^\infty z^2 j_\ell(qz) C_{\ell}\left(z, z^{\prime}\right) dz

=& \frac{\pi}{2} \left\{\int_0^\infty z'^2 dz' j_\ell(q'z') j_{\ell}(q z') \right\} P(q).\\

=& \frac{\pi}{2} \left\{ \frac{\pi}{2q'^2} \delta(q-q') \right\} P(q) \tag{4}.\\

\end{align}

To move the ##\delta## function in the right-hand-side, we multiply Eq. (4) (note that only ##q=q'## has contribution) with ##q'^2## and integral over ##q'##:

\begin{align}

\int_0^\infty dq' q'^2\int_0^\infty z'^2 dz' j_\ell(q'z') \int_0^\infty z^2 j_\ell(q'z) C_{\ell}\left(z, z'\right) dz

=& \frac{\pi^2}{4} \int_0^\infty dq' \delta(q-q') P(q).\\

=& \frac{\pi^2}{4} P(q) \tag{5}.

\end{align}

The left-hand-side of Eq.(5);

\begin{align}

\int_0^\infty dq' & q'^2\int_0^\infty z'^2 dz' j_\ell(q'z') \int_0^\infty z^2 j_\ell(q'z) C_{\ell}\left(z, z'\right) dz \\

= & \int_0^\infty z'^2 dz' \int_0^\infty z^2 dz \left\{ \int_0^\infty dq' q'^2 j_\ell(q'z') j_\ell(q'z) \right\} C_{\ell}(z, z') \\

= & \int_0^\infty z'^2 dz' \int_0^\infty z^2 dz \left\{ \frac{\pi}{2z^2} \delta(z-z') \right\} C_{\ell}(z, z') \\

= & \frac{\pi}{2} \int_0^\infty z^2 dz C_{\ell}(z, z). \tag{6}

\end{align}

Combine Eq.(5) and Eq.(6)

##

P(q) = \frac{2}{\pi} \int_0^\infty z^2 dz C_{\ell}(z, z).

##

in cosmology, the angular power spectrum depends on multipole noted π (Legendre transformation) which is related to angular quantities (π and π). But the matter power spectrum is dependent of π wave number (with Fourier transform).

I think I am wrong by saying that, in definition of πΆβ, one writes πΆβ(π§,π§β²) where π§ and π§β² could be understood like redshift.

**1) For example, I have the following demonstration,**##

C_{\ell}\left(z, z^{\prime}\right)=\int_{0}^{\infty} d k k^{2} j_{\ell}(k z) j_{\ell}\left(k z^{\prime}\right) P(k)

##

where ##j_{\ell}## are the spherical Bessel functions.

Given

## \tag{1}

C_{\ell}\left(z, z^{\prime}\right)=\int_{0}^{\infty} d k k^{2} j_{\ell}(k z) j_{\ell}\left(k z^{\prime}\right) P(k)

##

Question: how to invert the integral to find the function ##P(k)##?

==>

The closure relation for spherical Bessel function:

## \tag{2}

\int_0^\infty x^2 j_n(xu) j_n(xv) dx = \frac{\pi}{2u^2} \delta(u-v).

##

Multipy Eq.(1) with ##z^2 j_\ell(qz)## and integral over ##z##:

\begin{align}

\int_0^\infty z^2 j_\ell(qz) C_{\ell}\left(z, z^{\prime}\right) dz =&\int_{0}^{\infty} d k k^{2} \left\{ \int^0_\infty z^2 dz j_\ell(qz) j_{\ell}(k z)\right\} j_{\ell}\left(k z^{\prime}\right) P(k) \\

=&\int_{0}^{\infty} d k k^{2} \left\{\frac{\pi}{2q^2} \delta(q-k)\right\} j_{\ell}\left(k z^{\prime}\right) P(k) \\

=& q^{2} \frac{\pi}{2q^2} j_{\ell}\left(q z^{\prime}\right) P(q) \tag{3}.

\end{align}

Once again multiply Eq.(3) with ##z'^2 j_\ell(q'z')## and integral over ##z'##

\begin{align}

\int_0^\infty z'^2 dz' j_\ell(q'z') \int_0^\infty z^2 j_\ell(qz) C_{\ell}\left(z, z^{\prime}\right) dz

=& \frac{\pi}{2} \left\{\int_0^\infty z'^2 dz' j_\ell(q'z') j_{\ell}(q z') \right\} P(q).\\

=& \frac{\pi}{2} \left\{ \frac{\pi}{2q'^2} \delta(q-q') \right\} P(q) \tag{4}.\\

\end{align}

To move the ##\delta## function in the right-hand-side, we multiply Eq. (4) (note that only ##q=q'## has contribution) with ##q'^2## and integral over ##q'##:

\begin{align}

\int_0^\infty dq' q'^2\int_0^\infty z'^2 dz' j_\ell(q'z') \int_0^\infty z^2 j_\ell(q'z) C_{\ell}\left(z, z'\right) dz

=& \frac{\pi^2}{4} \int_0^\infty dq' \delta(q-q') P(q).\\

=& \frac{\pi^2}{4} P(q) \tag{5}.

\end{align}

The left-hand-side of Eq.(5);

\begin{align}

\int_0^\infty dq' & q'^2\int_0^\infty z'^2 dz' j_\ell(q'z') \int_0^\infty z^2 j_\ell(q'z) C_{\ell}\left(z, z'\right) dz \\

= & \int_0^\infty z'^2 dz' \int_0^\infty z^2 dz \left\{ \int_0^\infty dq' q'^2 j_\ell(q'z') j_\ell(q'z) \right\} C_{\ell}(z, z') \\

= & \int_0^\infty z'^2 dz' \int_0^\infty z^2 dz \left\{ \frac{\pi}{2z^2} \delta(z-z') \right\} C_{\ell}(z, z') \\

= & \frac{\pi}{2} \int_0^\infty z^2 dz C_{\ell}(z, z). \tag{6}

\end{align}

Combine Eq.(5) and Eq.(6)

##

P(q) = \frac{2}{\pi} \int_0^\infty z^2 dz C_{\ell}(z, z).

##

**2) I am surprized that ##C_{\ell}## has no dependence in π scale ? only angular dependent and redshift dependent ? since only redshift π§ appears in this expression ?**in cosmology, the angular power spectrum depends on multipole noted π (Legendre transformation) which is related to angular quantities (π and π). But the matter power spectrum is dependent of π wave number (with Fourier transform).

I think I am wrong by saying that, in definition of πΆβ, one writes πΆβ(π§,π§β²) where π§ and π§β² could be understood like redshift.

**But here, we talk about the ##C_{\ell}## of matter fluctuations and not temperature fluctuations, do you agree ?**

What do π§ and π§β² represent from your point of view in the expression πΆβ(π§,π§β²) ?

Where is my misunderstanding ?

Thanks in advance for your help and don't hesitate to ask me for further informations if I have not been clear enough.What do π§ and π§β² represent from your point of view in the expression πΆβ(π§,π§β²) ?

Where is my misunderstanding ?

Thanks in advance for your help and don't hesitate to ask me for further informations if I have not been clear enough.

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