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

We test the hypothesis that the expansion history of the universe can be described not by the standard ΛCDM model, but by a power-law scaling of the form a(t) ∝ t^α or even a dynamically evolving exponent α(t) modeled within a scalar–tensor framework.

Datasets Used

• Type Ia Supernovae (Pantheon+SH0ES)
  https://pantheonplussh0es.github.io/

• Gamma-Ray Bursts (GRB)
  Calibrated from Amati relation, see arXiv:1610.00854

• Cosmic Chronometers (CC)
  Hubble parameter from differential galaxy ages, see arXiv:2009.10701

• Planck 2018 CMB TT Power Spectrum
  Transformed into log-distance-like domain for joint likelihood fitting, see esdcdoi.esac.esa.int

---

Key Results

Dataset	Model	H₀	α or Ωₘ	χ²	AIC	BIC
CC	ΛCDM	68.16	0.319	14.55	18.55	21.49
	Time-Scaling	70.00	1.252	25.17	29.17	32.11
SN	ΛCDM	71.42	0.351	758.58	762.58	773.46
	Time-Scaling	70.89	1.388	763.42	767.42	778.30
GRB	ΛCDM	75.00	0.500	167.36	171.36	177.53
	Time-Scaling	75.00	1.000	174.97	178.97	185.15
CMB	ΛCDM	67.19	—	84136.8	84138.8	84144.6
	Time-Scaling	70.00	1.966	83939.5	83943.5	83955.1
Combined	ΛCDM	65.00	0.357	626542.7	626550.7	626576.3
	Time-Scaling	70.00	1.060	225288.4	225296.4	225321.9

ΔAIC = 401254.37 → Strong evidence in favor of Time-Scaling model in the global fit.

Scalar–Tensor Dynamics

We model α(t) as a scalar field coupled to gravity via a Brans–Dicke–like action, and test three potentials:

• Quadratic: V(α) ∝ α²
• Cosine: V(α) ∝ cos(α)
• Asymmetric: V(α) ∝ α³·sin(α)

Key finding: time-directionality emerges under asymmetric potentials — Lyapunov analysis shows:

• Forward evolution: λₗ ~ 10⁻³
• Backward evolution: λₗ < 0

No thermodynamic or quantum mechanisms are required to induce the arrow of time.

Repository Contents

.
├── model.py              # Full numerical analysis pipeline
├── results.txt           # Full fitting + MCMC + dynamics output
├── fig_obs.png           # Fits to SN, GRB, CC, and CMB
├── fig_obs_combined.png  # Combined observational fit
├── fig_dynamics.png      # Scalar field evolution (α(t))
├── fig_lyap.png          # Lyapunov exponent analysis
├── fig_mcmc.png          # Posterior samples for ΛCDM and Time-Scaling
└── README.md             # This file

How to Run

Requirements

Having Python ≥ 3.10, install dependencies:

pip install numpy scipy matplotlib pandas sympy emcee

Run the Pipeline

python model.py

Outputs include:

• Full observational fitting (SN, GRB, CC, CMB)
• Global AIC/BIC comparison
• Posterior sampling (optional MCMC)
• Scalar dynamics and Lyapunov evolution
• All plots saved as PNG

Citation

If using this code or results, please cite:

Chudzik, A. (2025). Revisiting the relationship between the scale factor a(t) and cosmic time t using numerical analysis. Mathematics (MDPI).