Curvature–matter couplings in modified gravity: From linear models to conformally invariant theories

In this paper, we review modified theories of gravity with a curvature–matter coupling between an arbitrary function of the scalar curvature and the Lagrangian density of matter. This explicit nonminimal coupling induces a nonvanishing covariant derivative of the energy–momentum tensor, that implies nongeodesic motion and consequently leads to the appearance of an extra force. Here, we explore the physical and cosmological implications of the nonconservation of the energy–momentum tensor by using the formalism of irreversible thermodynamics of open systems in the presence of matter creation/annihilation. The particle creation rates, pressure, and the expression of the comoving entropy are obtained in a covariant formulation and discussed in detail. Applied together with the gravitational field equations, the thermodynamics of open systems lead to a generalization of the standard [Formula: see text]CDM cosmological paradigm, in which the particle creation rates and pressures are effectively considered as components of the cosmological fluid energy–momentum tensor. Furthermore, we also briefly present the coupling of curvature to geometry in conformal quadratic Weyl gravity, by assuming a coupling term of the form [Formula: see text], where [Formula: see text] is the ordinary matter Lagrangian, and [Formula: see text] is the Weyl scalar. The coupling explicitly satisfies the requirement of the conformal invariance of the theory. Expressing [Formula: see text] with the use of an auxiliary scalar field and of the Weyl scalar, the gravitational action can be linearized in the Ricci scalar, leading in the Riemann space to a conformally invariant [Formula: see text]-type theory, with the matter Lagrangian nonminimally coupled to geometry.