Introduction

SzekeresPy is a Python package for cosmological calculations using the Szekeres Cosmological Model. The Szekeres model is an inhomogeneous, anisotropic, cosmological solution of Einstein's equations. In the limit of spherical symmetry it reduces to the Lemaitre-Tolman-Bondi (LTB) class of models, and in the limit of homogeneity, to FLRW models. The SzekeresPy can therefore be used to investigate LTB models (set dipole=0) or to perform calculations for FLRW cosmologies (set contrast=0), including the ΛCDM model.

SzekeresPy consists of two parts: a Python code for user interaction, and a Fortran code for handling the evolution and ray-tracing. Both the Fortran code and Python code are still in development. Debugging, optimisation, and restructuring of the code is still in progress. Anyone interested in collaboration or using the code is encouraged to contact the author.

Requirements

• Fortran compiler, e.g gfortran (the code was tested using GNU Fortran 11.4.0)
• Python 3.5 or higher (the code was tested using Python 3.10.12)
• numpy (the code was tested using numpy 1.21.5)
• files:
  • szekeres_fortran.f90
  • SzekeresPy.py
• matplotlib (for plotting)
• healpy (for sky maps)
• optional: astropy (for comparison and/or importing cosmological parameters)

Installation

To compile the code, navigate to the folder with the above files and run the terminal commands listed either in the file compile_single or compile_openmp (for openMP). The first step is to compile the fortran code and create a python module. For example, but typing:

python3 -m numpy.f2py -c --verbose --opt='-fopenmp -O3' -lgomp szekeres_fortran.f90 -m szekeres_fortran

you will compile the code (first "-c" in the line above) and create a python module (second "-m" in the command line above) which is then used by the SzekeresPy package. Then you can run the code from the terminal, by typing:

python3 sample.py

or you can use an integrated development environment, such as for example IDLE.

Running the code

Open the file sample.py and follow the sample calculations and examples presented in the file:

• Example 1: checking and updating cosmological parameters
• Example 2: using astropy to set the cosmological parameters
• Example 3: checking the properties of the Szekeres model at a given point (density, expansion rate, shear, Weyl)
• Example 4: plotting a 1d radial profile (for a specific direction) of: density, expansion rate, shear, Weyl
• Example 5: comparing radial density profile for two different directions
• Example 6: plotting a 2d density profile for a specific slice: x=const, or y=const, or z=const; (comoving domain only)
• Example 7: plotting a path of the light ray, t(R), featuring the apparent horizon (depending on the parameters of the model, somewhere around R ~ 2,000 Mpc - 5,000 Mpc).
• Example 8: plotting the angular diameter distance
• Example 9: plotting the luminosity distance
• Example 10: four sky maps: CMB, column density contrast, column expansion rate, shear (to run, set run_example_10 = True)

Development plans

• (削除) ver 0.1: evolution of a model (削除ここまで)
• (削除) ver 0.2: light propagation (削除ここまで)
• (削除) ver 0.3: distance-redshift relation (削除ここまで)
• (削除) ver 0.4: ray-tracing and sky maps (削除ここまで)
• ver 0.5: volume average with a variable position of the observer and radius of the averaging domain
• ver 0.6: dealing with mock data

Useful references

• https://arxiv.org/abs/astro-ph/0604490
• https://arxiv.org/abs/1412.4976
• https://arxiv.org/abs/1512.07364
• https://arxiv.org/abs/1704.02810
• https://arxiv.org/abs/2012.03192
• unpublished notes on light tracing in the Szekeres model (contact the author for a copy)

Questions and suggestions

In case of any questions, or suggestions for the code, please contact the author.