Generalized Chern-Pontryagin models

We formulate a new class of modified gravity models, that is, generalized four-dimensional Chern-Pontryagin models, whose action is characterized by an arbitrary function of the Ricci scalar R and the Chern-Pontryagin topological term ^*RR, i.e., f(R, ^*RR). Within this framework, we derive the gravitational field equations and solve them for a particular model, f(R, ^*RR)=R+β (^*RR)², considering two ansatzes: the slowly rotating metric and first-order perturbations of G\"{o}del-type metrics. For the former, we find a first-order correction to the frame-dragging effect boosted by the parameter L, which characterizes the departures from general relativity results. For the latter, G\"{o}del-type metrics hold unperturbed. We conclude this paper by displaying that generalized four-dimensional Chern-Pontryagin models admit a scalar-tensor representation, whose explicit form presents two scalar fields: Φ, a dynamical degree of freedom, while the second, ϑ, a non-dynamical degree of freedom. In particular, the scalar field ϑ emerges coupled with the Chern-Pontryagin topological term ^*RR, i.e., ϑ ^*RR, which is nothing more than Chern-Simons term.