Cosmological perturbations without the Boltzmann hierarchy

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 linearpolarization 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.

Automated summary

An alternative formulation for calculating the evolution of cosmological perturbations, involving solving two coupled integral equations to obtain the photon-intensity quadrupole, is proposed in this study. It provides some physical insight and serves as a cross-check for traditional methods, potentially accelerating current state-of-the-art cosmological-perturbation codes.