# Relationship between 3 quantities : Density Matter power spectrum, Density Matter angular power spectrum and Temperature angular power spectrum

I take over another post that is going to be deleted since I have none answer.

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 $$P(k)$$:

Maybe this is due to the fact that we talk about the $$C_{\ell}$$ of matter fluctuations and not temperature fluctuations (like in CMB angular power spectrum), could anyone confirm this ambiguity ?

1. For example, I have the following demonstration, $$C_{\ell}\left(z, z^{\prime}\right)=\int_{0}^{\infty} d k k^{2} j_{\ell}(k z) j_{\ell}\left(k z^{\prime}\right) P(k)\tag{1}$$ where $$j_{\ell}$$ are the spherical Bessel functions.

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

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

==> The closure relation for spherical Bessel function: $$\int_{0}^{\infty} x^{2} j_{n}(x u) j_{n}(x v) d x=\frac{\pi}{2 u^{2}} \delta(u-v)$$

Multipy $$(1)$$ with $$z^{2} j_{\ell}(q z)$$ and integral over $$z$$ :

\begin{aligned} \int_{0}^{\infty} z^{2} j_{\ell}(q z) C_{\ell}\left(z, z^{\prime}\right) d z &=\int_{0}^{\infty} d k k^{2}\left\{\int_{\infty}^{0} z^{2} d z j_{\ell}(q z) j_{\ell}(k z)\right\} j_{\ell}\left(k z^{\prime}\right) P(k) \\ &=\int_{0}^{\infty} d k k^{2}\left\{\frac{\pi}{2 q^{2}} \delta(q-k)\right\} j_{\ell}\left(k z^{\prime}\right) P(k) \\ &=q^{2} \frac{\pi}{2 q^{2}} j_{\ell}\left(q z^{\prime}\right) P(q)\quad(3) \end{aligned} Once again multiply $$(3)$$ with $$z^{\prime 2} j_{\ell}\left(q^{\prime} z^{\prime}\right)$$ and integral over $$z^{\prime}$$ \begin{aligned} \int_{0}^{\infty} z^{\prime 2} d z^{\prime} j_{\ell}\left(q^{\prime} z^{\prime}\right) \int_{0}^{\infty} z^{2} j_{\ell}(q z) C_{\ell}\left(z, z^{\prime}\right) d z &=\frac{\pi}{2}\left\{\int_{0}^{\infty} z^{\prime 2} d z^{\prime} j_{\ell}\left(q^{\prime} z^{\prime}\right) j_{\ell}\left(q z^{\prime}\right)\right\} P(q) \\ &=\frac{\pi}{2}\left\{\frac{\pi}{2 q^{\prime 2}} \delta\left(q-q^{\prime}\right)\right\} P(q) \quad(4) \end{aligned} To move the $$\delta$$ function in the right-hand-side, we multiply (4) (note that only $$q=q^{\prime}$$ has contribution) with $$q^{\prime 2}$$ and integral over $$q^{\prime}:$$ \begin{aligned} \int_{0}^{\infty} d q^{\prime} q^{\prime 2} \int_{0}^{\infty} z^{\prime 2} d z^{\prime} j_{\ell}\left(q^{\prime} z^{\prime}\right) \int_{0}^{\infty} z^{2} j_{\ell}\left(q^{\prime} z\right) C_{\ell}\left(z, z^{\prime}\right) d z &=\frac{\pi^{2}}{4} \int_{0}^{\infty} d q^{\prime} \delta\left(q-q^{\prime}\right) P(q) . \\ &=\frac{\pi^{2}}{4} P(q)\quad(5) \end{aligned}

The left-hand-side of Eq.(5); \begin{aligned} &\int_{0}^{\infty} d q^{\prime} q^{\prime 2} \int_{0}^{\infty} z^{\prime 2} d z^{\prime} j_{\ell}\left(q^{\prime} z^{\prime}\right) \int_{0}^{\infty} z^{2} j_{\ell}\left(q^{\prime} z\right) C_{\ell}\left(z, z^{\prime}\right) d z \\ &=\int_{0}^{\infty} z^{\prime 2} d z^{\prime} \int_{0}^{\infty} z^{2} d z\left\{\int_{0}^{\infty} d q^{\prime} q^{\prime 2} j_{\ell}\left(q^{\prime} z^{\prime}\right) j_{\ell}\left(q^{\prime} z\right)\right\} C_{\ell}\left(z, z^{\prime}\right) \\ &=\int_{0}^{\infty} z^{\prime 2} d z^{\prime} \int_{0}^{\infty} z^{2} d z\left\{\frac{\pi}{2 z^{2}} \delta\left(z-z^{\prime}\right)\right\} C_{\ell}\left(z, z^{\prime}\right) \\ &=\frac{\pi}{2} \int_{0}^{\infty} z^{2} d z C_{\ell}(z, z) . \end{aligned} \quad(6)

Combine (5) and (6) :

$$P(q)=\frac{2}{\pi} \int_{0}^{\infty} z^{2} d z C_{\ell}(z, z)$$

1. I am surprized that $$C_{\ell}$$ has no dependence in $$k$$ scale ?

Only angular dependent and redshift dependent ? since only redshift $$z$$ appears in this expression?

In cosmology, the angular power spectrum depends on multipole noted $$l$$ (Legendre transformation) which is related to angular quantities $$(\theta$$ and $$\phi)$$. But the matter power spectrum is dependent of $$k$$ wave number (with Fourier transform).

I think I am wrong by saying that, in definition of $$C \ell$$, one writes $$C \ell\left(z, z^{\prime}\right)$$ where $$z$$ and $$z$$' could be understood like redshift.

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

UPDATE :

What do $$z$$ and $$z^{\prime}$$ represent from your point of view in the expression $$C_\ell\left(z, z^{\prime}\right)$$ ?

It seems to correspond that $$C_\ell\left(z, z^{\prime}\right)$$ are elements of the covariance of $$C_\ell$$ with cross-correlations terms between redshift $$z$$ and $$z'$$, doesn't it ?

• This would be maybe more on-topic on Physics SE. – User123 Jun 16 at 14:16
• Isn't really there no one who could gives some clarifications ? – youpilat13 Jun 20 at 13:55
• I think that this site is maybe more for not-so-technical astronomy and the people on Physics SE could help you a little bit more. (But I believe that @ProfRob certainly has a solution to that.) – User123 Jun 20 at 19:41
• @User123 . Thanks for your advice but I posted the same kind of question on Physics SE and none answer. – youpilat13 Jun 20 at 20:31
• @ProfRob , Please if you could help me, even some elements of answer would be great ! Regards – youpilat13 Jun 20 at 20:32