The Einstein-Boltzmann equations revisited

TL;DR

This paper analyzes the mathematical properties of the Einstein-Boltzmann equations, focusing on stability, eigenvalues, and analytic solutions, to improve numerical cosmological simulations of universe perturbations.

Contribution

It provides a detailed eigenvalue and stability analysis of the Einstein-Boltzmann system, including regimes, analytic solutions, and criteria for numerical scheme stability.

Findings

Eigenvalue analysis reveals oscillation onset in the system.

Stability criteria for numerical schemes are established.

Analytic solutions are summarized for different regimes.

Abstract

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.