Unified description of cosmological and static solutions in affine generalized theories of gravity: Vecton–scalaron duality and its applications

We briefly describe the simplest class of affine theories of gravity in multidimensional space–times with symmetric connections and their reductions to two-dimensional dilaton–vecton gravity field theories. The distinctive feature of these theories is the presence of an absolutely neutral massive (or tachyonic) vector field (vecton) with an essentially nonlinear coupling to the dilaton gravity. We emphasize that the vecton field in dilaton–vecton gravity can be consistently replaced by a new effectively massive scalar field (scalaron) with an unusual coupling to the dilaton gravity. With this vecton–scalaron duality, we can use the methods and results of the standard dilaton gravity coupled to usual scalars in more complex dilaton–scalaron gravity theories equivalent to dilaton–vecton gravity. We present the dilaton–vecton gravity models derived by reductions of multidimensional affine theories and obtain one-dimensional dynamical systems simultaneously describing cosmological and static states in any gauge. Our approach is fully applicable to studying static and cosmological solutions in multidimensional theories and also in general one-dimensional dilaton–scalaron gravity models. We focus on general and global properties of the models, seeking integrals and analyzing the structure of the solution space. In integrable cases, it can be usefully visualized by drawing a "topological portrait" resembling the phase portraits of dynamical systems and simply exposing the global properties of static and cosmological solutions, including horizons, singularities, etc. For analytic approximations, we also propose an integral equation well suited for iterations.