Dynamics of quadratic f(R) cosmology with a perfect fluid: Jordan and Einstein frames

Abstract

We investigate the global dynamics of the field equations of (pure) quadratic theories of gravity which generalise Einstein's theory in spatially flat homogeneous and isotropic cosmological models with a perfect fluid. We introduce global and regular 3-dimensional dynamical systems' formulations, on both the Jordan frame and the conformally related Einstein frame. The analysis in the Jordan frame explores the monotonicity properties of the interior flow which, together with the characterisation of the orbit structure on the 2-dimensional invariant boundaries and the desingularisation of non-hyperbolic fixed points, provides a global description of the flow and its limit sets. In the Einstein frame, the analysis uses the skew-product structure of the Einstein state space and the characterisation of the orbit structure on the 2-dimensional invariant boundaries. Furthermore, by obtaining asymptotic expansions we identify the solutions that are global conformally mapped from the Jordan frame to the Einstein frame and those that are not.

Figures (24)

• Figure 1: The future cone state-space of the Jordan frame. The shaded region with $R<0$ corresponds to a conformal factor $F<0$.
• Figure 2: The global cylinder state-space in the Jordan frame. The shaded region corresponds to a conformal factor $F<0$.
• Figure 3: Successive blow-ups of $\mathrm{N}_0$ on $\{T=0\}$.
• Figure 4: The $\{T=0\}$ invariant boundary subset of $\bar{\bf S}_\mathrm{J}$.
• Figure 5: Blow-up of $\mathrm{N}_1$ on $\{T=1\}$.

Theorems & Definitions (36)

• Proposition 1
• proof
• Lemma 1
• proof
• Theorem 1
• proof
• Lemma 2: Blow-up of $\mathrm{N}_0$ on $\{T=0\}$
• proof
• Lemma 3: Orbit structure of $\{T=0\}$
• proof