The recollapse problem of closed Friedmann–Robertson–Walker models in higher-order gravity theories

Abstract

We study the closed universe recollapse conjecture for positively curved FRW models with a perfect fluid matter source and a scalar field which arises in the conformal frame of the R + αR 2 theory. By including ordinary matter, we extend the analysis of a previous work. We analyze the structure of the resulted fourdimensional dynamical system with the methods of the center manifold theory and the normal form theory. It is shown that an initially expanding closed FRW universe, starting close to the Minkowski spacetime, cannot avoid recollapse. We discuss the posibility that potentials with a positive minimum may prevent the recollapse of closed universes.

Oscillatory behaviour of closed isotropic models in second order gravity theory

“…System (21) is invariant under the transformation t → −t, y → −y (which implies that all trajectories are symmetric with respect to the r axis) and the line r = 0 is invariant. Note also that the system (21) has invariant lines y = ±2r (compare with the similar system in [41]). The behaviour of the solutions depends on the parameter γ and as we shall see, γ = 1 is a bifurcation value.…”

The Recollapse Problem of Closed Isotropic Models in Second Order Gravity Theory

“…For homogeneous and isotropic spacetimes in the Einstein frame this conjecture was found to be true for vacuum models [8] and for models with a perfect fluid with equation of state p = (γ − 1)ρ, 2/3 < γ < 2 [9]. More precisely it was shown that, an initially expanding closed FRW universe, starting close to the Minkowski spacetime cannot avoid recollapse.…”

“…Furthermore, the system (8) is not an arbitrary "free" four-dimensional system. In view of the 0 − 0 equation, the variables have to satisfy the constraint (9). The nonlinear coordinate transformation (14) is valid in a neighbourhood of the origin, small enough to ensure that the new variables still respect the constraint (9).…”

Some remarks about non-minimally coupled scalar field models

“…The zero minimum at φ = 0 of this potential is stable, but it cannot provide the mechanism for the late-time acceleration since V and H asymptotically approach to zero [156,157,192]. On the other hand, concerning the upper asymptote of V as φ → +∞, it follows that the potential has exponential order zero (see definition 2) as φ → +∞.…”

“…Preceding works for a large variety of non-negative potentials are [151,152,153,154,155,156,157,158]. In the reference [158], it have been extended many of the results obtained in [156] by considering arbitrary potentials.…”