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

In this proceeding, we review modified theories of gravity with a curvature-matter coupling between an arbitrary function the scalar curvature and Lagrangian density matter. This explicit nonminimal induces non-vanishing covariant derivative energy-momentum tensor, that implies non-geodesic motion consequently leads to appearance extra force. Here, explore physical cosmological implications nonconservation tensor by using formalism irreversible thermodynamics open systems in presence matter creation/annihilation. The particle creation rates, pressure, expression comoving entropy are obtained formulation discussed detail. Applied together gravitational field equations, lead generalization standard $\Lambda$CDM paradigm, which rates pressures effectively considered as components fluid tensor. Furthermore, also briefly present geometry conformal quadratic Weyl gravity, assuming term form $L_m\tilde{R}^2$, where $L_m$ is ordinary Lagrangian, $\tilde{R}$ scalar. explicitly satisfies requirement invariance theory. Expressing $\tilde{R}^2$ use auxiliary scalar, action can be linearized Ricci leading Riemann space conformally invariant $f\left(R,L_m\right)$ type theory, nonminimally coupled geometry.