Minimal Asymmetric Expansion of the Universe: A Friedmann-Sasaki-Randers Spacetime Framework Incorporating Dark Matter and Dark Energy

Abstract

In this work, we investigate the structure of the gravitational field and the Friedmann equations in a Sasaki–Randers spacetime, incorporating dark matter and dark energy within the Lorentz tangent bundle of spacetime. Additional terms arising in the geodesic equations and the Friedmann equations modify and generalize their standard forms in general relativity. Within this geometrical framework, the intuitive notion of distance as a symmetric and observer-independent quantity breaks down in relativistic and cosmological settings, where measurement procedures are time-extended, observer-dependent, and constrained by causal structure. Consequently, distance naturally acquires an asymmetric character, such that, in general, F(A,B)≠F(B,A). This intrinsic asymmetry admits a natural mathematical description in Finsler geometry, particularly through the Randers metric, where directional dependence introduces a preferred geometric flow. Within this framework, as developed in Finsler–Randers cosmology, anisotropy emerges not from matter distributions or initial conditions, but from the underlying structure of spacetime itself. The equivalence principle is also expanded with extra internal conditions on the tangent structure of spacetime and the Raychaudhuri equations are derived for this model. As a result, the universe appears observationally nearly isotropic, with only minimal anisotropies arising due to the direction-dependent nature of distance and cosmic expansion, while the standard notion of isotropy is recovered only as an approximation.