Modified gravity from Weyl connection and the f(R,cal{A}) extension

We use Weyl connection and Weyl geometry in order to construct novel modified gravitational theories. In the simplest case where one uses only the Weyl-connection Ricci scalar as a Lagrangian, the theory recovers general relativity. However, by upgrading the Weyl field to a dynamical field with a general potential and/or general couplings constructed from its trace, leads to new modified gravity theories, where the extra degrees of freedom arise from the Weyl field. Additionally, since the Weyl-connection Ricci scalar differs from the Levi-Civita Ricci scalar by terms up to first derivatives of the Weyl field, the resulting field equations for both the metric and the Weyl field are of second order, and thus the theory is free from Ostrogradsky ghosts. Finally, we construct the most general theory, namely the $f(\tilde{R},\cal{A})$ gravity, which is also ghost free. Applying the above classes of theories at a cosmological framework we obtain an effective dark energy sector of geometrical origin. In the simplest class of theories we are able to obtain an effective cosmological constant, and thus we recover $\Lambda$CDM paradigm, nevertheless in more general cases we acquire a dynamical dark energy. These theories can reproduce the thermal history of the Universe, and the corresponding dark energy equation-of-state parameter presents a rich behavior.