A Degenerate Bogdanov-Takens Normal Form for FLRW Cosmologies

TL;DR

This paper reformulates Einstein's equations for FLRW cosmologies as a degenerate Bogdanov-Takens normal form, revealing insights into solutions with constant expansion rate, negative curvature, and dust.

Contribution

It demonstrates that Einstein's equations near Minkowski spacetime can be expressed as a degenerate Bogdanov-Takens normal form, linking dynamical systems theory with cosmological solutions.

Findings

Normal form captures solutions with constant expansion and negative curvature.

Equivalence of Einstein equations to a normal form near Minkowski spacetime.

Identification of equilibrium points representing specific cosmological solutions.

Abstract

In this paper, we first show that the Einstein field equations for all perfect-fluid FLRW cosmologies can be written as a planar dynamical system with the equation of state parameter $w$ and cosmological constant $\Lambda$ as parameters. An important equilibrium point of this dynamical system is the origin which represents Minkowski spacetime. It is shown that the Einstein field equations in a neigbourhood of this point are equivalent to a degenerate Bogdanov-Takens normal form. This normal form admits a set of equilibrium points that describes a set of solutions to the Einstein field equations that have a constant rate of expansion, negative spatial curvature, zero cosmological constant and dust.