A numerical code for the solution of the Kompaneets equation in cosmological context

Context: After fundamental ground-based, balloon-born, and space experiments, and, in particular, after the COBE/FIRAS results, confirming that only very small deviations from a Planckian shape can be present in the CMB spectrum, current and future CMB absolute temperature experiments aim at discovering very small distortions such as those associated with the cosmological reionization process or that could be generated by different kinds of earlier processes. Aims: Interpretation of future data calls for a continuous improvement in the theoretical modeling of CMB spectrum. In this work we describe the fundamental approach and, in particular, the update to recent NAG versions of a numerical code, KYPRIX, specifically written to solve the Kompaneets equation in a cosmological context. It was first implemented in the years 1989-1991 to accurately compute the CMB spectral distortions under general assumptions. Methods: Specifically, we describe the structure and the main subdivisions of the code and discuss the most relevant aspects of its technical implementation. After a presentation of the equation formalism and of the boundary conditions added to the set of ordinary differential equations derived from the original parabolic partial differential equation, we provide details on the adopted space variable (i.e. dimensionless frequency) and space discretization, on time variables, on the output results, on the accuracy parameters, and on the used auxiliary integration routines. The problem with introducing the time dependence of the ratio between electron and photon temperatures and of the radiative Compton scattering term, both of them introducing integral terms into the Kompaneets equation, is addressed in the specific context of the recent NAG versions. We describe the introduction of the cosmological constant in the terms controlling the general expansion of the Universe in agreement with the current concordance model, of the relevant chemical abundances, and on the ionization history, from recombination to cosmological reionization. The global computational time, the impact of the various aspects of the code on it, and the accuracy of the numerical integration are also discussed. Results: We present some of fundamental tests we carried out to verify the accuracy, reliability, and performance of the code. We focus on some quantitative tests of energy conservation and the time behavior of electron temperature. A comparison of the results obtained with the update and the original version of the code is presented for some representative cases. Finally, we focus on some properties of the free-free distortions relevant for the long wavelength region of the CMB spectrum. Conclusions: All the tests demonstrate the reliability and versatility of the new code version and its accuracy and applicability to the scientific analysis of current CMB spectrum data and of much more precise measurements that will be available in the future. The recipes and tests described in this work can also be useful for implementing accurate numerical codes for other scientific purposes using the same or similar numerical libraries or for verifying the validity of different codes aimed at the same problem or similar ones.