The Evolution of the Universe Based on Principal Bundle Geometry

Abstract

This paper proposes a novel geometric framework for unifying dark matter, dark energy, and cosmic evolution through a modified gauge-theoretic approach. By introducing a scalar function γ ( x ) = tanh ( k x ) , where x = a a 0 (with a as the local “absolute acceleration” and a 0 ≈ 10 − 8 cm/s² a reference scale), we construct an extended principal bundle P ˜ = P × ℝ and redefine the connection form as ω ˜ = ω G + d γ , ensuring curvature invariance while modulating gravitational sources. In the Newtonian limit, the field equation simplifies to γ ∇ 2 ϕ = − 4 π ρ , revealing γ ’s role in scaling gravitational effects. Distinctively, dark matter and dark energy emerge as geometric consequences of spacetime curvature rather than exotic components: γ → 1 restores standard gravity at small scales, 0 < γ < 1 enhances gravity (mimicking dark matter) at galactic scales, and γ → 0 + drives cosmic acceleration via Λ ( γ ) → ∞ . Crucially, allowing x to take negative values introduces γ → 0 − with Λ ( γ ) → − ∞ , enabling cyclic universe scenarios through curvature oscillations. The derived force law F = γ m a aligns with MOND phenomenology, bridging modified dynamics and geometric modulation. Furthermore, the model constructs a closed “Taiji-like” cosmic evolution loop in the γ − x plane, linking asymptotic states ( γ = ± 1 at x = ± ∞ ) via hyperbolic perturbations, symbolizing cyclic rebirth. By unifying cosmic dynamics under a geometric gauge framework, this work offers testable predictions for parameter calibration ( k , a 0 , Λ 0 ) and establishes a philosophically resonant synthesis of mathematical physics and cosmological evolution.