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

TL;DR

This paper explores the explicit integration of Friedmann's equations with nonlinear equations of state, using Chebyshev's theorem and other methods, to analyze various cosmological models and their physical implications.

Contribution

It extends the integrability analysis of Friedmann equations to nonlinear equations of state, providing explicit solutions and insights into cosmological parameters' roles.

Findings

Some models are integrable via Chebyshev's theorem.

Explicit solutions reveal the impact of nonlinear matter on dark matter.

Applications extend beyond cosmology to physics and geometry.

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.