Oscillatory behaviour of closed isotropic models in second order gravity theory

Abstract

Homogeneous and isotropic models are studied in the Jordan frame of the second order gravity theory. The late time evolution of the models is analysed with the methods of the dynamical systems. The normal form of the dynamical system has periodic solutions for a large set of initial conditions. This implies that an initially expanding closed isotropic universe may exhibit oscillatory behaviour.

Comment on “Existence of Einstein static universes and their stability in fourth-order theories of gravity”

“…In view of the above interpretation, it is easy to check that substituting fðRÞ ¼ R þ R 2 in (1) with à ¼ 0, the scalar curvature takes the constant value given by (3), exactly as was shown in Ref. [3].…”

“…However, in a previous investigation it was found that this is not the case [3]. More precisely, without making use of expansionnormalized variables it is shown in Ref.…”

“…More precisely, without making use of expansionnormalized variables it is shown in Ref. [3] that the dynamical system has an unstable equilibrium of saddle type corresponding to the Einstein static universe. What is most interesting is that the effective cosmological constant is not assumed from the beginning, but is provided by the curvature equilibrium…”

On stability of the Kasner solution in quadratic gravity

“…Moreover, Kasner solutions are "building blocks" for BKL chaotic approach to a singularity in more general models like Bianchi IX mode [40]l. The study of asymptotic solutions in quadratic gravity was previously addressed, for example, by [41][42][43][44].…”

Asymptotic solutions in f(R)-gravity

“…If the equation of state parameter γ < 1, the smooth part of solution coincides with GR behaviour, otherwise the influence of quadratic curvature correction becomes important at late times (see [35,36]). …”