Relationship between the angular and 3D power spectra

[John321]

TL;DR Summary

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

I have the following equation,
$$ C_\ell(z,z') = \int_0^\infty dkk^2 j_\ell(kz)j_\ell(kz')P(k),$$

where $$j_\ell$$ are the spherical Bessel functions.

I would like to invert this relation and write P(k) as a function of C_l. I don't know if this is a well known result, but I couldn't find anything. I've also looked into Fredholm integral equations, but that got me nowhere, and now I'm stuck.

Any ideas on how to tackle this problem?

 

[AndyW]

Hello,

Thank you for sharing your equation with us. This is a challenging problem, but I believe there are a few approaches you can take to try and solve it.

One option is to use numerical methods to approximate the inverse relation. This would involve discretizing the integral and using numerical integration techniques to solve for P(k). However, this may not be the most efficient or accurate approach.

Another option is to use mathematical techniques such as Laplace transforms or Fourier transforms to manipulate the equation and potentially find a closed-form solution for P(k). This could involve using properties of the Bessel functions or other known integral identities.

You mentioned looking into Fredholm integral equations, which can be a useful tool for solving inverse problems. However, in this case, it may not be the most appropriate approach as it typically involves solving for unknown functions in integral equations with a known kernel, whereas in your equation, both the kernel and the unknown function are unknown.

It may also be helpful to consult with colleagues or experts in the field to see if they have any insights or suggestions for solving this type of problem. I hope this gives you some ideas on how to approach this problem. Good luck with your research!