Revisiting the Relationship Between the Scale Factor ( a ( t )) and Cosmic Time ( t ) Using Numerical Analysis

Background: Current cosmological fits typically assume a direct relation between cosmic time ( t ) and the scale factor ( a ( t ) ), yet this ansatz remains largely untested across diverse observations. Objectives: We (i) test whether a single power-law scaling ( a ( t ) ∝ t α ) can reproduce late- and early-time cosmological data and (ii) explore whether a dynamically evolving ( α ( t ) ), modeled as a scalar–tensor field, naturally induces directional asymmetry in cosmic evolution. Methods: We fit a constant- α model to four independent datasets: 1701 Pantheon+SH0ES supernovae, 162 gamma-ray bursts, 32 cosmic chronometers, and the Planck 2018 TT spectrum (2507 points). The CMB angular spectrum is mapped onto a logarithmic distance-like scale ( μ = log 10 D ℓ ), allowing for unified likelihood analysis. Each dataset yields slightly different preferred values for H 0 and α ; therefore, we also perform a global combined fit. For scalar–tensor dynamics, we integrate α ( t ) under three potentials—quadratic, cosine, and parity breaking ( α 3 sin α )—and quantify directionality via forward/backward evolution and Lyapunov exponents. Results: (1) The constant- α model achieves good fits across all datasets. In combined analysis, it yields H 0 ≃ 70 km s − 1 Mpc − 1 and α ≃ 1.06 , outperforming Λ CDM globally ( Δ AIC ≃ 401254 ), though Λ CDM remains favored for some low-redshift chronometer data. High-redshift GRB and CMB data drive the improved fit. Numerical likelihood evaluations are approximately three times faster than for Λ CDM. (2) Dynamical α ( t ) models exhibit time-directional behavior: under asymmetric potentials, forward evolution displays finite Lyapunov exponents ( λ L ∼ 10 − 3 ), while backward trajectories remain confined ( λ L < 0 ), realizing classical arrow-of-time emergence without entropy or quantum input. Limitations: This study addresses only homogeneous background evolution; perturbations and physical derivations of potentials remain open questions. Conclusions: The time-scaling approach offers a computationally efficient control scenario in cosmological model testing. Scalar–tensor extensions naturally introduce classical time asymmetry that is numerically accessible and observationally testable within current datasets. Code and full data are available.