Non-integrability of density perturbations in the FRW universe

TL;DR

This paper proves that the linear density perturbation equations in a standard cosmological model are generally not solvable in closed form, except for some known solutions, using Kovacic's algorithm.

Contribution

It applies Kovacic's algorithm to demonstrate the non-integrability of the density perturbation equations in the FRW universe.

Findings

No closed-form solutions exist beyond known special cases.

The analysis confirms the equations' non-integrability.

Potential for more complex solutions involving special functions is limited.

Abstract

We investigate the evolution equation of linear density perturbations in the Friedmann-Robertson-Walker universe with matter, radiation and the cosmological constant. The concept of solvability by quadratures is defined and used to prove that there are no "closed form" solutions except for the known Chernin, Heath, Meszaros and simple degenerate ones. The analysis is performed applying Kovacic's algorithm. The possibility of the existence of other, more general solutions involving special functions is also investigated.