On asymptotic solutions of Friedmann equations

Abstract

Our main aim is to apply the theory of regularly varying functions to the asymptotical analysis at infinity of solutions of Friedmann cosmological equations. A new constant $\Gamma$ is introduced related to the Friedmann cosmological equations. Determining the values of $\Gamma$ we obtain the asymptotical behavior of the solutions, i.e. of the expansion scale factor $a(t)$ of a universe. The instance $\Gamma < 1/4$ is appropriate for both cases, the spatially flat and open universe, and gives a sufficient and necessary condition for the solutions to be regularly varying. This property of Friedmann equations is formulated as the generalized power law principle. From the theory of regular variation it follows that the solutions under usual assumptions include a multiplicative term which is a slowly varying function.