Emergence of the Gambier equation in cosmology

Abstract

We show how the Gambier equation arises in connection to Friedmann-Lema$\mbox{î}$tre-Robertson-Walker (FLRW) cosmology and a Dark Matter equation of state. Moreover, we provide a correspondence between the Friedmann equations and the Gambier equations that possess the Painlev$\acute{\mbox{e}}$ property in $2+1$ dimensions. We also consider special cases of the Gambier G27 equation such as the generalized Pinney equation. For an extended FLRW model with dynamic scalar field as matter model, the Einstein equations correspond to the Milne-Pinney equation which in turn can be mapped to the parametric Gambier equation of second order.

arXiv comment13 pages

References in corpus (13)

• The Black Hole in Three Dimensional Space Time
• Geometry of the 2+1 Black Hole
• Inhomogeneous Equation of State of the Universe: Phantom Era, Future Singularity and Crossing the Phantom Barrier
• The Ermakov-Pinney Equation in Scalar Field Cosmologies
• Is the future universe singular: Dark Matter versus modified gravity?
• Cosmological aspects of the Eisenhart-Duval lift
• Cosmological scaling solutions of minimally coupled scalar fields in three dimensions
• Cosmic Dynamics of Bose-Einstein Condensates
• Dark energy in some integrable and nonintegrable FRW cosmological models
• Painlevé structure of a multi-ion electrodiffusion system
• From Feshbach-Resonance Managed Bose-Einstein Condensates to Anisotropic Universes: Some Applications of the Ermakov-Pinney equation with Time-Dependent Nonlinearity
• Universal procedure to cure future singularities of dark energy models
• Non-linear Schrödinger-type formulation of scalar field cosmology: two barotropic fluids and exact solutions