Introduction to Modified Gravity and Gravitational Alternative for Dark Energy

TL;DR

This review surveys a broad class of modified gravity theories as gravitational alternatives to dark energy, focusing on $f(R)$, $f(G)$, $f(R,G)$, and Gauss–Bonnet–dilaton constructions that can drive late-time acceleration and even phantom-like behavior without exotic matter. It highlights the scalar-tensor equivalence, frame dependencies, and conditions for phantom divide crossing, as well as exact solutions that realize acceleration while remaining compatible with Solar System tests in certain regimes. The discussion extends to non-linear curvature–matter couplings, dynamical cosmological constant scenarios, and unified GB-based approaches that may realize inflation and late-time acceleration, along with the prospect of inhomogeneous EoS dark fluids arising from modified gravity. The work also analyzes singularity structure and the origins of inhomogeneous terms, underscoring the need for further theoretical development and observational constraints to identify viable, testable models of cosmic acceleration.

Abstract

We review various modified gravities considered as gravitational alternative for dark energy. Specifically, we consider the versions of $f(R)$, $f(G)$ or $f(R,G)$ gravity, model with non-linear gravitational coupling or string-inspired model with Gauss-Bonnet-dilaton coupling in the late universe where they lead to cosmic speed-up. It is shown that some of such theories may pass the Solar System tests. On the same time, it is demonstrated that they have quite rich cosmological structure: they may naturally describe the effective (cosmological constant, quintessence or phantom) late-time era with a possible transition from decceleration to acceleration thanks to gravitational terms which increase with scalar curvature decrease. The possibility to explain the coincidence problem as the manifestation of the universe expansion in such models is mentioned. The late (phantom or quintessence) universe filled with dark fluid with inhomogeneous equation of state (where inhomogeneous terms are originated from the modified gravity) is also described.