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 the particular models, \(f(R, ^*RR)=R+\beta ( ^*RR)^2\) and \(f(R, ^*RR)=R+\alpha R^2+\beta ( ^*RR)^2\), considering two ansatzes: the slowly rotating Schwarzschild metric and first-order perturbations of Gödel-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, Gödel-type metrics hold unperturbed, for specific sort of perturbed metric functions. 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: \(\Phi \), a dynamical degree of freedom, while the second, \(\vartheta \), a non-dynamical degree of freedom. In particular, the scalar field \(\vartheta \) emerges coupled with the Chern–Pontryagin topological term \( ^*RR\), i.e., \(\vartheta ^*RR\), which is nothing more than Chern–Simons term.