The Einstein-Boltzmann equations revisited
  The linear Einstein-Boltzmann 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 signaled 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 Einstein-Boltzmann 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 wave number $k$ and scale factor $a$. We also
provide a compendium of analytic solutions for all perturbation variables in 6
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.
