Dynamic Fractal Cosmology: A Fibonacci Phase Transition Model

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Overview v1.3.0

This repository contains the LaTeX source and associated files for the research paper titled "Dynamic Fractal Cosmology: A Fibonacci Phase Transition Model." This work introduces a complete fractal cosmological framework where the golden ratio $\phi$ evolves dynamically from primordial ($\phi_0=1.5$) to modern ($\phi_\infty=1.618$) epochs. This phase transition, characterized by a rate parameter $\mathbf{\Gamma=0.23\pm0.01}$, resolves the Hubble tension and explains Cosmic Microwave Background (CMB) anomalies through scale-dependent fractal dimensions.

Crucially, leveraging Pantheon+ Type Ia supernova data, our model yields a best-fit Hubble constant of $\mathbf{H_0=72.82\pm0.16 \text{ km/s/Mpc}}$, along with $\mathbf{\Omega_m=0.270\pm0.065}$ and an absolute magnitude $\mathbf{M=-19.38\pm0.16 \text{ mag}}$, demonstrating an an excellent fit with $\mathbf{\chi^2/\text{dof}=0.61}$.

The model makes several key predictions:

1. BAO Deviations: $\Delta r_d/r_d \approx 0.15(1-e^{-z/2})$.
2. CMB Power Deficit: $\mathcal{S}=0.93\pm0.02$ at $\ell<30$, yielding a significantly better $\chi^2/\text{dof}=1.72$ compared to $5.40$ for a static fractal model with constant $\phi=1.5$ (using Planck 2018 TT+lowE data).
3. Redshift-dependent Growth: $f(z)=\Omega_m(z)^{\phi(z)/2}$.

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Repository Contents

This repository is structured as follows:

• main.tex: The primary LaTeX source file for the manuscript.
• main.pdf: The compiled PDF version of the research paper.
• references.bib: The BibTeX file containing all references cited in the manuscript.
• .zenodo.json: A configuration file for Zenodo, specifying metadata for automated archiving and DOI assignment.
• CITATION.cff: A Citation File Format file, providing citation metadata for this repository.
• LICENSE.md: A CC BY 4.0 license.
• figures/: Directory containing all figures included in the paper.
  • figures/hubble_diagram_comparison.pdf: Hubble Diagram showing the distance modulus vs redshift, comparing the Dynamic Fractal Model with SNIa data and $\Lambda$CDM.
  • figures/hz_comparison_plot.pdf: Evolution of the relative Hubble parameter $H(z)/H_0$ vs redshift, for the Dynamic Fractal Model and $\Lambda$CDM.
• scripts/: (If applicable) Directory intended for any analysis or data fitting scripts.
  • scripts/snia_dynamic_fractal_analysis.py: (If applicable) Python script used for fitting the model to Pantheon+ Supernovae Type Ia data.

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Key Findings and Visualizations

The research highlights several critical aspects of the dynamic fractal cosmology model, supported by illustrative figures within the paper:

• Figure 1: Evolution of the fractal dimension Shows the transition between primordial ($\phi_0 = 1.5$) and modern ($\phi_\infty = 1.618$) values, reflecting the $\Gamma$ parameter derived from SNIa data.
• Figure 2: Convergence of Fibonacci ratios toward $\phi$ Illustrates how the primordial value $\phi_0 = 1.5$ ($F_4/F_3$) marks the onset of fractal dimensionality.
• Figure 3: CMB spectrum showing fractal corrections at $\ell<30$ (blue band) Compares the model's predictions to $\Lambda$CDM (dashed line) and Planck 2018 data points.
• Figure 4: Hubble constant measurements with $1\sigma$ errors Presents the model's best-fit $H_0$ from Pantheon+ SNIa data, demonstrating how it reconciles local and CMB measurements.
• Figure 5: Hubble Diagram Comparison Illustrates the agreement between the Dynamic Fractal Model and Pantheon+ SNIa data, showing a superior fit compared to the standard $\Lambda$CDM model.
• Figure 6: $H(z)/H_0$ vs Redshift Comparison Displays the evolution of the relative Hubble parameter, highlighting the higher relative expansion rate at low redshifts predicted by our model, consistent with local $H_0$ measurements.
• Figure 7: Evolution of Energy Densities and Fractal Transition Compares the evolution of matter and dark energy densities in the Dynamic Fractal Model versus $\Lambda$CDM, emphasizing the impact of the $\phi(z)$ transition.

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Compilation Requirements

To compile the main.tex document, the following LaTeX packages are utilized:

• inputenc (utf8)
• fontenc (T1)
• amsmath, amssymb
• graphicx
• natbib
• tikz
• pgfplots (with compat=1.18 set)
• tikzlibrary{shapes.geometric, arrows.meta, calc}
• booktabs
• xcolor
• hyperref
• orcidlink
• ragged2e
• textcomp

A standard LaTeX distribution (e.g., TeX Live, MiKTeX) that includes these packages is sufficient for compilation.

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Citation

If you utilize this work, please cite the corresponding Zenodo deposit using the Concept DOI, which always points to the latest version:

You can also find the full citation details in the CITATION.cff file.

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License

This project is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). You are free to share and adapt the material for any purpose, even commercially, provided you give appropriate credit, provide a link to the license, and indicate if changes were made.

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Contact

For any inquiries or further information regarding this research, please contact:

Sylvain Herbin

• ORCID: https://orcid.org/0009-0001-3390-5012
• Email: herbinsylvain@protonmail.com
• Website: https://sylvainherbin.github.io/cosmo/