Cosmological perturbations without the Boltzmann hierarchy

TL;DR

This paper introduces an alternative approach to calculating cosmological perturbations that bypasses the traditional Boltzmann hierarchy by solving coupled integral equations for photon quadrupoles, offering efficiency and physical insight.

Contribution

The paper presents a new integral-equation-based formulation for cosmological perturbations that reduces computational complexity and enhances physical understanding.

Findings

The method provides a faster numerical solution for cosmological perturbations.

It offers a cross-check for the traditional Boltzmann hierarchy approach.

The approach may accelerate existing cosmological-perturbation codes.

Abstract

Calculations of the evolution of cosmological perturbations generally involve solution of a large number of coupled differential equations to describe the evolution of the multipole moments of the distribution of photon intensities and polarization. However, this "Boltzmann hierarchy" communicates with the rest of the system of equations for the other perturbation variables only through the photon-intensity quadrupole moment. Here I develop an alternative formulation wherein this photon-intensity quadrupole is obtained via solution of two coupled integral equations -- one for the intensity quadrupole and another for the linear-polarization quadrupole -- rather than the full Boltzmann hierarchy. This alternative method of calculation provides some physical insight and a cross-check for the traditional approach. I describe a simple and efficient iterative numerical solution that converges fairly quickly. I surmise that this may allow current state-of-the-art cosmological-perturbation codes to be accelerated.