Coupling matter and curvature in Weyl geometry- conformally invariant f(R,L_m) gravity

We propose an extension of the $f(R,L_m)$ gravity theory by considering the coupling between matter and geometry in conformal quadratic Weyl gravity, explicitly formulated in the Weyl geometry. In order to keep the conformal invariance of the theory we add in the gravitational action a coupling term of the form $L_m\tilde{R}^2$, where $L_m$ is the ordinary matter Lagrangian, and $\tilde{R}$ is the Weyl scalar. The gravitational action constructed in this way explicitly is conformally invariant. We linearize the theory by introducing an auxiliary scalar field, which allows to formulate the gravitational action in the Riemann geometry. The gravitational field equations of the theory, as well as the energy-momentum balance equations are obtained by varying the action with respect to the metric. The divergence of the matter energy-momentum tensor does not vanish, and an extra force, depending on the Weyl vector, and on the matter Lagrangian, does appear, leading to a nongeodesic motion of massive test particles. The generalized Poisson equation is also obtained, and the Newtonian limit of the equations of motion is considered in detail. The perihelion precession of the planet Mercury is used to obtain some observational constraints on the magnitude of the Weyl vector in the Solar System. The cosmological implications of the theory are investigated for the case of a flat, homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker geometry. The cosmological model can give a good description of the observational data for the Hubble function up to a redshift of the order of $z\approx 3$.