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

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.

💡 Research Summary

This paper presents a comprehensive global dynamical analysis of pure quadratic f(R)=αR² gravity coupled to a perfect fluid with a linear equation of state (γ∈(2/3,2)) in spatially flat Friedmann‑Robertson‑Walker (FRW) cosmologies. The authors treat both the Jordan frame, where the higher‑order curvature term appears explicitly, and the conformally related Einstein frame, where the theory is recast as General Relativity with a minimally coupled scalar field ϕ and a fluid that interacts with ϕ through a fixed coupling β=√(2/3).

Jordan‑frame formulation.
Starting from the action S_J=∫