Class of viable modified $f(R)$ gravities describing inflation and the onset of accelerated expansion
G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani, S. Zerbini
TL;DR
This work develops a general framework for viable modified gravity using $f(R)$ theories in both the Jordan and Einstein frames, emphasizing exponential and step-like models that interpolate between high-curvature inflation and low-curvature late-time acceleration. It establishes viability criteria (e.g., $f(0)=0$, large-$R$ behavior, and positive scalaron mass) and analyzes stability via de Sitter critical points and a dynamical-systems approach, including conditions $eta(R_*)>1$. The authors construct explicit one- and two-step exponential models, assess their solar-system consistency through a heavy scalaron and negligible Newtonian corrections, and discuss smoothing to avoid antigravity, with extensions to $N$-step constructions. They also translate these models to the Einstein frame to study the scalar potential $V(\sigma)$ and the coupled dynamics, providing explicit forms and discussing implications for PPN/cosmology, laying groundwork for data-driven constraints on unified inflation and acceleration scenarios.
Abstract
A general approach to viable modified $f(R)$ gravity is developed in both the Jordan and the Einstein frames. A class of exponential, realistic modified gravities is introduced and investigated with care. Special focus is made on step-class models, most promising from the phenomenological viewpoint and which provide a natural way to classify all viable modified gravities. One- and two-steps models are explicitly considered, but the analysis is extensible to $N$-step models. Both inflation in the early universe and the onset of recent accelerated expansion arise in these models in a natural, unified way. Moreover, it is demonstrated that models in this category easily pass all local tests, including stability of spherical body solution, non-violation of Newton's law, and generation of a very heavy positive mass for the additional scalar degree of freedom.