Mijajlović Ž.  

Applications of regularly varying functions in the study of cosmological parameters

Most of the cosmological parameters, such as the scale factor a(t), the energy density ρ(t) and p(t), the pressure of the material in the universe,  satisfy asymptotically the power law. On the other hand the quantities that satisfy the power law are best modeled by regularly varying functions. The aim of this paper is to apply the theory of regularly varying functions to study Friedmann equations and their solutions which are in fact the mentioned cosmological parameters. More specifically, in our paper On asymptotic solutions of Friedmann equations (Ž. Mijajlović, N. Pejović, S. Šegan, and G. Damljanović, Applied Mathematics and Computation, 2012) we introduced a new constant Γ related to the Friedmann equations. Determining the values of Γ we obtain the asymptotical behavior of the  solutions, i.e. of the expansion scale factor a(t). The instance Γ < 1/4 is appropriate for both cases, the spatially flat and the 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.