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

TL;DR

This paper develops a unified framework for describing cosmological and static solutions in affine generalized gravity theories, introducing vecton-scalaron duality to simplify complex models and analyze their global properties.

Contribution

It introduces a vecton-scalaron duality in affine gravity theories, enabling the use of scalar models to study complex vector-scalar coupled systems in cosmology and static solutions.

Findings

Derived one-dimensional dynamical systems for cosmological and static states.

Established a duality allowing vector fields to be replaced by scalar fields in models.

Provided methods for analyzing global properties and solution spaces, including horizons and singularities.

Abstract

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 (DVG). The distinctive feature of these theories is the presence of an absolutely neutral massive (or tachyonic) vector field (vecton) with essentially nonlinear coupling to the dilaton gravity (DG). We show that in DVG the vecton field can be consistently replaced by an effectively massive scalar field (scalaron) with an unusual coupling to dilaton gravity. With this vecton - scalaron duality, one can use methods and results of the standard DG coupled to usual scalars (DGS) in more complex dilaton - scalaron gravity theories (DSG) equivalent to DVG. We present the DVG 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 as well as in general one-dimensional DGS models. We focus on global properties of the models, look for integrals and analyze the structure of the solution spaces. In integrable cases, it can be usefully visualized by drawing a `topological portrait' resembling phase portraits of dynamical systems and simply exposing global properties of static and cosmological solutions, including horizons, singularities, etc. For analytic approximations we also propose an integral equation well suited for iterations.