Explicit Integration of Friedmann's Equation with Nonlinear Equations of State

Abstract

This paper is a continuation of our earlier study on the integrability of the Friedmann equations in the light of the Chebyshev theorem. Our main focus will be on a series of important, yet not previously touched, problems when the equation of state for the perfect-fluid universe is nonlinear. These include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born--Infeld, and two-fluid models. We show that some of these may be integrated using Chebyshev's result while other are out of reach by the theorem but may be integrated explicitly by other methods. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution. For example, in the Chaplygin gas universe, it is seen that, as far as there is a tiny presence of nonlinear matter, linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.

43 pages no figures

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• Geometrothermodynamics
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• Einstein-Maxwell-Dilatonic phantom black holes: Thermodynamics and geometrothermodynamics
• Infinite dimensional symmetry groups of the Friedmann equations
• Dyonic Matter Equations, Exact Point-Source Solutions, and Charged Black Holes in Generalized Born--Infeld Theory
• Analogy between freezing lakes and the cosmic radiation era
• Cosmological analogies, Lagrangians, and symmetries for convective-radiative heat transfer
• Lagrangian formulation of Omoriis law and analogy with the cosmic Big Rip