Friedmann-Lemaitre Cosmologies via Roulettes and Other Analytic Methods

TL;DR

This paper introduces novel analytic methods, including roulette representations and asymptotic analysis, to solve and understand complex Friedmann equations in cosmology, revealing new phenomena and universal growth laws.

Contribution

It develops new methods for solving Friedmann equations beyond Chebyshev theorem limitations, including roulette representations and asymptotic analysis, applied to various cosmological models.

Findings

Representation of solutions as roulettes enables new insights.

Discovery of rich phenomena in quadratic and Randall--Sundrum models.

Universal exponential growth law in complex equations of state.

Abstract

In this work a series of methods are developed for understanding the Friedmann equation when it is beyond the reach of the Chebyshev theorem. First it will be demonstrated that every solution of the Friedmann equation admits a representation as a roulette such that information on the latter may be used to obtain that for the former. Next the Friedmann equation is integrated for a quadratic equation of state and for the Randall--Sundrum II universe, leading to a harvest of a rich collection of new interesting phenomena. Finally an analytic method is used to isolate the asymptotic behavior of the solutions of the Friedmann equation, when the equation of state is of an extended form which renders the integration impossible, and to establish a universal exponential growth law.