Accelerating universes generated by off-diagonal deformations and geometric flows of black holes in Einstein gravity vs $f(R)$ gravity

Abstract: Over the past two decades, numerous modifications of general relativity (GR) and the standard cosmological paradigm have been proposed to describe both the early inflationary epoch and the late - time accelerating expansion of the Universe, while remaining consistent with updated observational data. In this work, we argue that it is possible to effectively stop the machinery of producing new modified gravity theories and exotic cosmological models, and instead remain within an axiomatic, spacetime-geometric framework of GR closely related to the standard ΛCDM paradigm. To support this claim, we construct new classes of generic off-diagonal solutions in GR and in the relativistic geometric flow theory of nonholonomic Einstein systems. These solutions describe the geometric evolution of black hole configurations into accelerating cosmological universes with effective dark energy fluids. Using the anholonomic frame and connection deformation method, we decouple and integrate, exactly or parametrically, the underlying nonlinear field equations for general off-diagonal metrics and (non)linear connection distortions. The resulting configurations, characterized by generating functions, integration functions, and effective sources, exhibit nonlinear symmetries and running cosmological constants, allowing smooth transformations between black hole and cosmological geometries.