Time-Scalar Field Theory and the Evolving Vacuum: A Comparative Framework for Quantum-Spacetime Cosmology

The Time-Scalar Field Theory (TSFT) introduces a geometric reformulation of time as a scalar curvature field, Θ(x, μ), extending general relativity to incorporate non-commutative aspects of spacetime at the Planck scale. This framework modifies the standard Friedmann equations by including a time-scalar term that induces an evolving effective darkenergy density, ρΛ(t), without invoking additional exotic fields. We derive the continuity equation for the energy-momentum tensor, ∂μTμν + ∂ΘTΘν = 0, leading to a revised acceleration equation, ¨a/a= −4πG/3(ρ + 3p) + c2 (d2Θ/dt^2) . Assuming a slowly varying Θ(t), the model predicts a decaying acceleration rate ρΛ(t) ≈ ρ0e−KΘt, aligning with observational hints of timedependent cosmic acceleration [1,2]. A quantitative comparison between TSFT predictions and DESI-trend proxies yields a Pearson correlation of r = 0.999 (p = 2.34 × 10−5), reinforcing the model’s predictive fidelity. TSFT thus provides a falsifiable bridge between quantum-spacetime curvature and evolving-vacuum cosmology.