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 Γ is introduced related to the Friedmann cosmological equations. Determining the values of Γ we obtain the asymptotical behavior of the solutions, i.e. of the expansion scale factor a(t) of a universe. The instance Γ < 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.

Asymptotic solution for expanding universe with matter-dominated evolution

“…The significance of the integral limit (1.14) is seen in the following facts. We have shown in Mijajlović et al [27] that for any function µ ∈ M such that Γ µ < 1/4, the Friedmann equations has a normalized regularly varying solutions and the universe modelled by these solutions must be spatially flat or open. On the other hand, if Γ µ > 1/4 then the universe is oscillatory.…”

“…The function M(µ) is a linear functional M : M → R, where M is the space of all real functions satisfying the above integral condition (1.14). M is called a Marić class of functions (see Mijajlović et al [27]). If this integral limit does not exist, we say that Γ does not exist.…”

“…If this integral limit does not exist, we say that Γ does not exist. The constant Γ appears in description of cosmological parameters by use of the theory of regularly varying solutions of linear second order differential equations, see Mijajlović et al [27], Mijajlović, Pejović & Marić [28] and Marić [25]. Determining the values of Γ one can obtain the asymptotical behaviour of the solutions a(t), ρ(t) and p(t).…”

“…In the introduction we present some basic facts on Friedmann equations, short history and elements of regular variation needed in the rest of the paper. Also, properties of the constant Γ introduced in Mijajlović et al [27] are reviewed. This constant and the related operator M will have to play the crucial role in our analysis of asymptotics of cosmological parameters.…”

“…There were also some uses of this theory in cosmology, particularly in the study of asymptotic behaviour of cosmological parameters, e.g. Mijajlović et al [27], Mijajlović, Pejović & Marić [28], but also by Molchanov, Surgailis & Woyczynski [29], Stern [37]. Barrow [6] and Barrow & Shaw [7] had a similar approach in studies of asymptotic behaviour of solutions to the Einstein equations describing expanding universes.…”