Fourier and Hankel analysis

Two files in the folder tests (namely test_fft.py and test_fftlog.py) show some brief and documented example of how to use the functions contained in Fourier and Hankel transforms. Alternatively, in the folder notebooks the equivalent Python notebooks can be found. Here we just want to show how to use the FFTlog-based functions colibri.fourier.iFFT_iso_3D() function to switch from Fourier space and compute correlation functions in real space.

We know that the two-point correlation function is the 3D Fourier transform of the power spectrum, i.e.

\[\xi(r) = \int \frac{d^3\mathbf{k}}{(2\pi)^3} \ P(k) \ e^{i \mathbf{k\cdot r}} = \int_0^\infty \frac{dk \ k^2}{2\pi^2} \ P(k) \ \frac{\sin kr}{kr}\]

where the second equality follows from statistical isotropy of the Universe. Mathematically, the equation above is similar to an inverse Hankel transform of order 1/2 (with some coefficients in front).

To compute that, we just need a few lines

import numpy as np
import colibri.cosmology as cc
import matplotlib.pyplot as plt
import colibri.fourier as FF

C = cc.cosmo()

# Linear matter power spectrum
k, pk = C.camb_Pk(z = [0., 1., 2])

# Fourier transforms for each redshift
xi = []
for i in range(len(pk)):
    r_tmp, xi_tmp = FF.iFFT_iso_3D(k, pk[i])
    xi.append(xi_tmp)

# Save array of correlation functions
r  = r_tmp
xi = np.array(xi)

Here is the outcome: the BAO feature is clearly visible.