Hamiltonian Dynamics of Cosmological Quintessence Models

TL;DR

This paper analyzes the Hamiltonian dynamics of two cosmological quintessence models based on real gases, revealing stable cyclic solutions and explicit trajectories, thus deepening understanding of their evolution.

Contribution

It introduces a Hamiltonian framework for nonlinear cosmological gas models, identifying conserved quantities and explicit solutions, advancing the analysis of cyclic universe scenarios.

Findings

Existence of stable periodic solutions indicating cyclic universes

Explicit solutions for certain trajectories in the models

Static equilibria are only reachable at infinite time

Abstract

The time-evolution dynamics of two nonlinear cosmological real gas models has been reexamined in detail with methods from the theory of Hamiltonian dynamical systems. These examples are FRWL cosmologies, one based on a gas, satisfying the van der Waals equation and another one based on the virial expansion gas equation. The cosmological variables used are the expansion rate, given by the Hubble parameter, and the energy density. The analysis is aided by the existence of global first integral as well as several special (second) integrals in each case. In addition, the global first integral can serve as a Hamiltonian for a canonical Hamiltonian formulation of the evolution equations. The conserved quantities lead to the existence of stable periodic solutions (closed orbits) which are models of a cyclic Universe. The second integrals allow for explicit solutions as functions of time on some special trajectories and thus for a deeper understanding of the underlying physics. In particular, it is shown that any possible static equilibrium is reachable only for infinite time.