Double power series method for approximating cosmological perturbations

TL;DR

This paper introduces a double power series method based on the FSN approach to analytically approximate solutions for cosmological perturbations, achieving high accuracy and providing a complete set of sub-horizon solutions.

Contribution

It adapts the FSN double power series method to cosmological perturbation theory, offering a new analytical approximation technique for sub-horizon scales.

Findings

Achieves ~1% accuracy compared to numerical solutions.

Provides a complete set of oscillating and non-oscillating solutions.

Includes a Mathematica implementation for practical use.

Abstract

We introduce a double power series method for finding approximate analytical solutions for systems of differential equations commonly found in cosmological perturbation theory. The method was set out, in a non-cosmological context, by Feshchenko, Shkil' and Nikolenko (FSN) in 1966, and is applicable to cases where perturbations are on sub-horizon scales. The FSN method is essentially an extension of the well known Wentzel-Kramers-Brillouin (WKB) method for finding approximate analytical solutions for ordinary differential equations. The FSN method we use is applicable well beyond perturbation theory to solve systems of ordinary differential equations, linear in the derivatives, that also depend on a small parameter, which here we take to be related to the inverse wave-number. We use the FSN method to find new approximate oscillating solutions in linear order cosmological perturbation theory for a flat radiation-matter universe. Together with this model's well known growing and decaying M\'esz\'aros solutions, these oscillating modes provide a complete set of sub-horizon approximations for the metric potential, radiation and matter perturbations. Comparison with numerical solutions of the perturbation equations shows that our approximations can be made accurate to within a typical error of 1%, or better. We also set out a heuristic method for error estimation. A Mathematica notebook which implements the double power series method is made available online.