Modified Gravity as an Alternative to Dark Energy: A Survey of Curvature, Torsion, Non-metricity and Matter Geometry Coupling Theories

S. R. Bhoyar, A. D. Wankhade

Abstract: The observed late-time acceleration of the Universe remains one of the deepest open problems in fundamental physics. Within general relativity it demands a dark energy component most economically a cosmological constant Λ whose observationally inferred value is separated from naive quantum field theoretic expectations by tens of orders of magnitude. Modified theories of gravity offer an alternative: cosmic acceleration as a manifestation of infrared modifications of the gravitational sector itself. This review surveys the principal geometric routes to such modifications, organized by the geometric trinity of gravity: curvature-based theories (fR , scalar–tensor and Horndeski, Gauss–Bonnet extensions), torsion-based theories (fT and generalizations), and non-metricity-based theories (fQ and generalizations), together with the increasingly studied class of matter–geometry coupling theories, fR,T , fR,Lm , fQ,T , fQ,Lm and energy–momentum-squared gravity. We apply three viability filters uniformly across all classes: (i) local and astrophysical constraints, foremost the multimessenger bound cT/c-1≲10-15 from GW170817; (ii) consistency with late-time cosmological observations and the status of the H0 and S8 tensions; and (iii) thermodynamic viability, i.e. the generalized second law and non-negative entropy production at the apparent horizon, a criterion that can discriminate between models degenerate at the background level and that acquires particular significance in matter–geometry coupling theories, where non-conservation of the energy–momentum tensor is naturally interpreted as irreversible gravitationally induced particle creation. We summarize the theoretical pathologies afflicting each class, delineate the surviving theory space, and outline the discriminating power of forthcoming surveys and gravitational-wave observations.