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 of the scalar curvature
and the Lagrangian density of matter. This explicit nonminimal coupling induces
a non-vanishing covariant derivative of the energy-momentum tensor, that
implies non-geodesic 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
$\Lambda$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 $L_m\tilde{R}^2$, where $L_m$ is the ordinary matter
Lagrangian, and $\tilde{R}$ is the Weyl scalar. The coupling explicitly
satisfies the requirement of the conformal invariance of the theory. Expressing
$\tilde{R}^2$ 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 $f\left(R,L_m\right)$ type theory,
with the matter Lagrangian nonminimally coupled to geometry.
