Calculating the non-Limber expression for the angular power spectrum using the Levin integration.

The non-Limber integral which is computed for each element of the angular power spectrum is, for clustering:

for cosmic shear:

and for galaxy-galaxy lensing (assuming quantities labeled with '1' are associated with shear and '2' with number counts):

where K is in each case the kernel for the appropriate tracer and is the non-linear matter power spectrum. We assume that .

Compiling

The code is compiled by typing make install. In the main directory there is another makefile for an example main. The documentation can be created via doxygen -g doxyfile followed by doxygen doxyfile . Only standard libraries and the GSL are needed.

For the python module, pip install . (or pip install -e . for a develop install) should do the trick. An up-to-date version of pip (10+) is required. You might have to install pybind11, but this should work automatically.

Running the code

There exists a python test file showing which things one should pass to the constructor of the class. In principle the class needs to be initialized once. This is especially true if the Bessel functions are precomputed since this would generate unnecessary overhead. If you want to update the spectra, the weights or anything, you should just call the init_splines method to set up the new splines without touching the Bessel functions.

The arguments in the constructor are:

• precompute: if set True, Besselfunctions are precomputed, speeds up the calculation.
• number_count: the first number_count entries are treated as the galaxy clustering, while the remaining are treated as cosmic shear.
• z_bg and chi_bg: two arrays of the same length specifying the relation between redshift and comoving distance
• chi_cl and kernel: this specifies the Integration Kernels, all kernels must have the same comoving distance range. kernel itself is an array with len(chi_cl) rows. The number of columns corresponds to the number of kernels included and is the sum of the number_count and the number of cosmic shear bins.
• k_pk, z_pk and pk: the matter power spectrum over which is integrated (all bias terms are must be included in the kernel). pk is a one-dimensional array and must have k as the fast index and z as the slow one. (see example)

The output of compute_C_ells is a list of the galaxy-clustering, galaxy-galaxy lensing and cosmic shear spectra respectively. The individual spectra are ordered under the condition i<=j with respect to tomographic bin combination (see also test_levin.py)