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In this work a series of methods are developed for understanding theFriedmann equation when it is beyond the reach of the Chebyshev theorem. Firstit will be demonstrated that every solution of the Friedmann equation admits arepresentation as a roulette such that information on the latter may be used toobtain that for the former. Next the Friedmann equation is integrated for aquadratic equation of state and for the Randall--Sundrum II universe, leadingto a harvest of a rich collection of new interesting phenomena. Finally ananalytic method is used to isolate the asymptotic behavior of the solutions ofthe Friedmann equation, when the equation of state is of an extended form whichrenders the integration impossible, and to establish a universal exponentialgrowth law.

Friedmann-Lemaitre Cosmologies via Roulettes and Other Analytic Methods