The Einstein-Boltzmann equations revisited

Abstract

The linear Einstein–Boltzmann (E-B) equations describe the evolution of perturbations in the universe and its numerical solutions play a central role in cosmology. We revisit this system of differential equations and present a detailed investigation of its mathematical properties. For this purpose, we focus on a simplified set of equations aimed at describing the broad features of the matter power spectrum. We first perform an eigenvalue analysis and study the onset of oscillations in the system signalled by the transition from real to complex eigenvalues. We then provide a stability criterion of different numerical schemes for this linear system and estimate the associated step size. We elucidate the stiffness property of the E-B system and show how it can be characterized in terms of the eigenvalues. While the parameters of the system are time dependent making it non-autonomous, we define an adiabatic regime where the parameters vary slowly enough for the system to be quasi-autonomous. We summarize the different regimes of the system for these different criteria as function of wavenumber k and scalefactor a. We also provide a compendium of analytic solutions for all perturbation variables in six limits on the k–a plane and express them explicitly in terms of initial conditions. These results are aimed to help the further development and testing of numerical cosmological Boltzmann solvers.