cl_to_xi_wigner_d

Compute correlation functions from power spectra

File	shear/cl_to_xi_wigner/wigner_cl_xi_interface.py
Attribution	Joe Zuntz
	Nicolas Tessore
URL
Citations	https://arxiv.org/abs/1904.09973

This module creates and stores transformation matrices to convert power spectra to correlation functions.

There is a general relation [arXiv:1904.09973]

\(\xi(\theta) = \sum_\ell (2*\ell + 1)/(4 \pi) C_\ell d^\ell_{ss'}(\theta)\)

to convert between cl and xi of spin s and s’ fields, using the reduced Wigner d-functions. This is a linear relation that can be written in matrix form. Once the values of theta are fixed, the matrix never changes.

A complication is that the theory Cls are not computed for all integer l up to some ell_max, so that the available values must be interpolated at the integers. Since linear interpolation is also a linear operation, it can be absorbed directly into the transformation matrix, which as a result has size n_theta x n_ell even if the above sum stretches to ell_max >> n_ell. The interpolation is currently done linearly in ell, but could perhaps be changed to be log-linear for better results.

The transformation matrices are computed on the first evaluation, which takes a long time if ell_max is high, since (ell_max+1) x n_theta Wigner d-function values are calculated. However, after the first time, each cl-to-xi conversion is a cheap n_theta x n_ell matrix multiplication, so one might come out ahead over a long run. These matrices could in principle even be stored to disk for given settings of ell and theta.

Assumptions

• Input C_ell up to large ell_max 10_000 ~ 100_000.

Setup Parameters

Name	Type	Default	Description
corr_type	int or str		Type of the correlation function: 0/xi, 1/wtheta, 2/gammat
input_section_name	str	shear_cl, galaxy_cl, or galaxy_shear_cl, depending on corr_type	Name of the angular power spectrum input section. See shear/spectra module. (Default: )
output_section_name	str	shear_xi, galaxy_shear_xi, or galaxy_xi depending on corr_type	Name of the angular correlation function output section

Input values

Section	Name	Type	Default	Description
input_section_name	ell	real 1d		Sample ell values for input C_ell
	nbin_a	int		The number of redshift bins in the first quantity
	nbin_b	int		The number of redshift bins in the second quantity
	nbin	int		Number of redshift bins used if nbin_a or b not found.
	bin_i_j	real 1d		C_ell (no l(l+1) factor) for bin i and j. Only need j<=i for modes 0 and 1.

Output values

Output values

Section	Name	Type	Description
output_section_name	theta	real 1d	Sample theta values for output xi(theta)
	bin_i_j	real 1d	w(theta) or gamma_t(theta) for bin i and j, only if mode=1 or 2 respectively. Only stores j<=i.
	nbin_a	int	Number of i tomographic bins
	nbin_b	int	Number of j tomographic bins