Fibonacci Cosmology: Falsified Background, Testable Perturbations

Author: Bryan David Persaud
Affiliation: Intermedia Communications Corp.
Contact: bryan@imediacorp.com

---

Overview

This repository accompanies two complementary works investigating whether the Golden Ratio (φ ≈ 1.618) manifests in cosmology:

1. Background model (falsified): tests φ-recursion in the expansion history. Result: decisively ruled out by H(z), CMB low-ℓ, and P(k) data.
2. Perturbation model (falsifiable): proposes weak log-periodic modulations in structure growth, yielding φ-spaced features in the matter power spectrum and CMB. Provides concrete forecasts for near-future surveys.

The code here reproduces the core figures, basic fits, and forecast visuals, and lets you explore signals using public summary datasets.

Note on Independence: This cosmological research work is completely independent and separate from any other projects. It originated as a simple thought experiment and hypothesis that the Golden Ratio (Fibonacci sequence) might be fundamental to cosmic structure, given the self-similarity patterns observed from plants to galaxies. This work has no connection or linkage to any other Fibonacci-related research projects or modules.

Theorem overview (short)

• Golden Ratio as recursive constant: The hypothesis explores φ as a fundamental recursion constant. In a background variant (now falsified), φ enters the expansion rate via recursive scaling; in the perturbation variant, φ induces log-periodic modulations in clustering statistics.
• Duality for perturbations: Two coupled modes appear naturally — a forward (φ) and a conjugate (φ−1) branch — leading to a dual log-periodic signature. In practice, this means residuals may contain cosines periodic in log(k) or log(ℓ) with periods set by ln(φ) and ln(φ−1).
• Falsifiable prediction: If these dual log-periodic features are absent at the sensitivity of current/forthcoming surveys, the perturbation hypothesis is ruled out.

---

Papers

• fibonacci_cosmology_falsified.pdf — Background model and its observational falsification
• fibonacci_perturbations.pdf — Perturbation-level hypothesis and detectability forecasts

If you use this repository, please cite the corresponding paper(s).

Paper summaries

• Background (falsified): A φ-recursive background expansion was tested against observations. H(z) fits, low-ℓ CMB residuals, and matter P(k) collectively rule it out with large Δχ². Key takeaway: φ is not a viable background constant for cosmic acceleration.
• Perturbations (falsifiable): Keeping ΛCDM background, introduce dual log-periodic modulations governed by φ and φ−1 in the growth of structure. Predicts φ-spaced features in P(k) around the BAO scale and small CMB residual oscillations. Forecasts suggest near-future surveys could detect or refute these signals decisively.

---

Repository Contents

New φ-Modulation Analysis Framework

A scientifically defensible framework implementing a two-parameter extension to ΛCDM testing for log-periodic oscillations in the matter power spectrum at scales determined by the golden ratio φ. This represents a shift from hypothesis-proving to hypothesis-testing with proper statistical rigor.

New components:

• src/phi_modulation.py — Core PhiModulationModel class with CAMB integration
• notebooks/01_desi_forecasts.ipynb — DESI Year 5 forecast analysis with Fisher matrix methodology
• figures/desi_forecasts.png — 4-panel forecast figure (generated by notebook)

Quick start with new framework:

from src.phi_modulation import PhiModulationModel

# Initialize model with Planck 2018 parameters
model = PhiModulationModel()

# Generate base ΛCDM power spectrum
k, z, Pk = model.get_base_power_spectrum(k_min=0.01, k_max=0.3)

# Apply φ-modulation: P_mod = P_ΛCDM × [1 + A_φ × cos(2π × log(k/k_pivot) / ln(φ) + φ_0)]
Pk_mod, mod_factor = model.apply_phi_modulation(k, Pk[0], A_phi=0.01)

# Forecast DESI sensitivity using Fisher matrix
result = model.forecast_desi_sensitivity(A_phi_true=0.01)
print(f"Forecast σ_Aφ = {result['sigma_Aphi']:.4f}, SNR = {result['SNR']:.2f}σ")

Run the forecast notebook:

jupyter notebook notebooks/01_desi_forecasts.ipynb

Legacy Scripts and Notebooks

Key scripts and notebooks (legacy analysis):

• hz_fitter_v2.py — Fit H(z) cosmic-chronometer data against φ-recursive model and ΛCDM
• cmb_osc_detector_v2.py — Search for log-periodic residuals in low-ℓ CMB TT
• lss_phi_analyzer.py — Analyze matter power spectrum P(k) for φ-spaced oscillatory residuals
• forecast_plots.py — Minimal example producing φ-oscillation forecast plot(s)
• fetch_desi_bao_pk.py — Fetch/approximate DESI BAO summary and synthesize a sample P(k)
• fibonacci_cosmology_analysis.ipynb — Notebook version of analysis/figures

Data (CSV):

• real_hz.csv — H(z) measurements (z, H, σ_H)
• real_cmb_lowl.csv — Planck low-ℓ CMB TT (ℓ, C_ℓ, σ)
• real_pk_lowk.csv — Low‑k matter power spectrum (k, P(k), σ)
• pk_data.csv / pk_data_real.csv — Example/synthesized P(k) data (from scripts)

Figures (generated):

• hz_comparison_*.png — H(z) comparison plots
• pk_phi*.png — P(k) residual/feature plots
• cmb_osc*.png — CMB residual log‑periodic fit plots
• *_forecast.png — Forecast visuals

Docs:

• FibonacciCosmologyAnalysis.md, CosmologicalTestResultsAnalysis.md, CosmicPerturbationHypothesis.md — narrative notes
• LicenseAgreement.md — license and usage terms

---

Usage Guide

New φ-Modulation Framework

Run the DESI forecast analysis notebook:

jupyter notebook notebooks/01_desi_forecasts.ipynb

This notebook:

• Tests different modulation amplitudes (A_φ = 0.005, 0.01, 0.02)
• Generates 4-panel forecast figure:
  • Power spectrum ratio (P_mod/P_base - 1) vs k
  • BAO signature (ξ(r) comparison)
  • DESI SNR vs amplitude
  • Comparison with existing constraints
• Prints forecast summary with σ_Aφ and SNR for each amplitude

Use the PhiModulationModel class in your own code:

from src.phi_modulation import PhiModulationModel
import numpy as np

# Initialize with default Planck 2018 parameters
model = PhiModulationModel()

# Or specify custom parameters
custom_params = {
    'H0': 67.36,
    'ombh2': 0.02237,
    'omch2': 0.1200,
    'As': 2.1e-9,
    'ns': 0.9649,
    'tau': 0.0544
}
model = PhiModulationModel(params=custom_params)

# Generate ΛCDM power spectrum
k, z, Pk = model.get_base_power_spectrum(k_min=0.01, k_max=0.3, npoints=500)

# Apply φ-modulation
A_phi = 0.01  # Amplitude
phi_phase = 0.0  # Phase offset
k_pivot = 0.05  # Pivot scale [h/Mpc]
Pk_mod, mod_factor = model.apply_phi_modulation(
    k, Pk[0], A_phi=A_phi, phi_phase=phi_phase, k_pivot=k_pivot
)

# Compute BAO signature (correlation function)
r, xi_base, xi_mod = model.compute_bao_signature(z=0.5, A_phi=A_phi)

# Forecast DESI sensitivity
forecast = model.forecast_desi_sensitivity(A_phi_true=A_phi)
print(f"Forecast uncertainty: σ_Aφ = {forecast['sigma_Aphi']:.4f}")
print(f"Signal-to-noise: SNR = {forecast['SNR']:.2f}σ")

Legacy Scripts

• Run hz_fitter_v2.py for H(z) falsification of the φ-recursive background against ΛCDM:

  python hz_fitter_v2.py

• Run cmb_osc_detector_v2.py for CMB low-ℓ residual log-periodic fit (dual φ/φ−1 modes):

  python cmb_osc_detector_v2.py

• Run lss_phi_analyzer.py for φ-spaced residuals in the matter power spectrum P(k):

  python lss_phi_analyzer.py

• Run forecast_plots.py for simple φ-oscillation forecast visualization:

  python forecast_plots.py

• Optional: run fetch_desi_bao_pk.py to reconstruct a sample P(k) from DESI DR2 BAO summary and save pk_data_real.csv:

  python fetch_desi_bao_pk.py

---

Data descriptions

• real_hz.csv: Cosmic-chronometer H(z) catalog (columns: z, H, sigma_H).
• real_cmb_lowl.csv: Planck TT low-ℓ summary (columns: ell, C_ell, sigma).
• real_pk_lowk.csv: SDSS/BOSS P(k) for low-k scales (columns: k, P(k), sigma_Pk).
• pk_data.csv: Simple synthetic P(k) used by examples.
• pk_data_real.csv: Reconstructed P(k) produced by fetch_desi_bao_pk.py (DESI DR2–like).

---

Requirements

• Python 3.9–3.12
• Packages (see requirements.txt):
  • numpy, scipy, matplotlib, pandas
  • astropy (for ΛCDM baseline comparisons)
  • requests, beautifulsoup4 (for fetching)
  • sympy, pillow (utilities/figures)
  • camb (required for new φ-modulation framework — generates ΛCDM power spectra)
  • emcee, dynesty, corner (for Bayesian analysis — optional but recommended)
  • pyccl, desilike (optional — advanced cosmology tools)

Install:

python -m venv .venv && source .venv/bin/activate  # on Windows: .venv\\Scripts\\activate
pip install -r requirements.txt
# If astropy is missing in your environment:
pip install astropy

---

Quick Start (GitHub/local)

Reproduce the main figures using the included CSVs. All scripts write PNG outputs to the project root by default.

1. H(z) comparison (falsification figure):

python hz_fitter_v2.py

Outputs: hz_comparison_dual.png and console χ² values.

2. CMB low‑ℓ residual log‑periodic fit:

python cmb_osc_detector_v2.py

Outputs: cmb_osc_dual.png and fitted amplitudes/phases (console).

3. Matter power spectrum residuals and φ‑scales:

python lss_phi_analyzer.py

Outputs: pk_phi.png and console summary of peak matches.

4. Forecast visual for P(k):

python forecast_plots.py

Outputs: pk_forecast.png

5. Optional: synthesize P(k) from DESI BAO summary, then analyze:

python fetch_desi_bao_pk.py   # creates pk_data_real.csv
python lss_phi_analyzer.py    # analyzes the CSV already present (or adjust path inside)

Notes:

• If internet is disabled or the DESI URL changes, the fetch script falls back to a small DR2‑like mock table and still produces pk_data_real.csv.

---

Using on Kaggle

Kaggle Notebooks tips:

• Kaggle link: https://www.kaggle.com/bryanpersaud
• FAAC Notebook: https://www.kaggle.com/code/bryanpersaud/faac-notebook/
• New: fibonacci_cosmology_kaggle.py — a Kaggle-ready script-notebook with robust data loading (GitHub online and local CSV offline fallback), quick-look plots, and a toy log-periodic fit. Add this file to your Kaggle Notebook (as a script) and run cells top-to-bottom.
• Internet: Turn Internet on if you want fetch_desi_bao_pk.py to download BAO data or to load CSVs directly from GitHub. If off, upload real_hz.csv, real_cmb_lowl.csv, and real_pk_lowk.csv as Kaggle Dataset files; the loader falls back to local.
• Data: The repository includes sample CSVs under the root; ensure they’re available in your Kaggle working directory (Dataset upload or Git clone step).
• Environment: Use a Python 3.10+ image. Install deps at the top of your notebook:

!pip -q install numpy scipy matplotlib pandas astropy sympy pillow requests beautifulsoup4

• Running: Use the same commands as above in notebook cells (prefix with !), or simply run fibonacci_cosmology_kaggle.py cells to reproduce the quick-look analyses.
• Outputs: Figures are saved to the working directory; use Kaggle’s “Output Files” to view/download PNGs.
• Background vs Perturbations: See docs/fib_recursion.md for the revised write-up aligning with this repo (background falsified; perturbations testable).

---

Run in Google Colab

If you prefer Colab, start a new notebook, then run:

• Install dependencies (as requested):

!pip install -q numpy scipy matplotlib pandas astropy camb astroquery emcee corner

• Clone or upload this repo/files, then run the same commands as locally, e.g.:

!python hz_fitter_v2.py
!python cmb_osc_detector_v2.py
!python lss_phi_analyzer.py

Notes:

• Some optional packages (e.g., camb, astroquery, emcee, corner) are not strictly required by the included scripts but are useful for advanced experimentation.
• Figures will be written to /content by default; use the Files panel to download.

---

Reproducibility & Notes

• Scripts are deterministic given the fixed input CSVs; random seeds are not used.
• Baselines: Some routines approximate ΛCDM trends (e.g., polynomial baseline for low‑ℓ CMB) to isolate residuals; these are analysis conveniences, not full Boltzmann codes.
• Units: P(k) is plotted in customary units; axes are labeled in each figure; see script headers for details.

---

Project Structure

.
├── src/                          # New: Core analysis modules
│   ├── __init__.py
│   └── phi_modulation.py        # PhiModulationModel class
├── notebooks/                    # New: Analysis notebooks
│   └── 01_desi_forecasts.ipynb  # DESI forecast analysis
├── figures/                      # New: Output figures
│   └── desi_forecasts.png       # Generated by notebook
├── forecasts/                    # New: Forecast-specific outputs
├── docs/                         # Documentation and notes
│   └── New Fibonacci Test Theorem.md  # New approach documentation
├── fibonacci_cosmology_falsified.pdf
├── fibonacci_perturbations.pdf
├── real_hz.csv
├── real_cmb_lowl.csv
├── real_pk_lowk.csv
├── pk_data_real.csv
├── hz_fitter_v2.py              # Legacy: H(z) analysis
├── cmb_osc_detector_v2.py       # Legacy: CMB analysis
├── lss_phi_analyzer.py          # Legacy: P(k) analysis
├── forecast_plots.py            # Legacy: Forecast plots
├── fetch_desi_bao_pk.py         # DESI data fetching
├── fibonacci_cosmology_analysis.ipynb  # Legacy notebook
└── requirements.txt

---

Data Sources & Attribution

• H(z) cosmic chronometers: aggregated public compilations as provided in real_hz.csv (see file header/commit history for source notes).
• Planck low-ℓ TT: simplified summary exported to real_cmb_lowl.csv for residual analysis convenience; refer to Planck Collaboration papers for the authoritative spectra.
• Low‑k P(k): real_pk_lowk.csv represents a reduced P(k) sample suitable for demonstrating analysis workflows; see comments and commit history for provenance.
• DESI DR2 BAO summary: fetched from data.desi.lbl.gov when Internet is enabled; otherwise fetch_desi_bao_pk.py uses a small offline fallback with DR2‑like parameters described in-script.

Please credit the original survey teams when appropriate (Planck Collaboration, SDSS/BOSS/eBOSS, DESI Collaboration). This repository provides lightweight derivatives for method demonstration.

---

License

See LicenseAgreement.md for terms. If you need a conventional SPDX identifier for packaging, contact the author to clarify redistribution terms.

---

Citation

If this work contributes to your research, please cite the appropriate paper and this repository. Example placeholder (update with final bibliographic data):

@misc{persaud_fibonacci_cosmology,
  author       = {Bryan David Persaud},
  title        = {Fibonacci Cosmology: Falsified Background, Testable Perturbations},
  year         = {2025},
  howpublished = {GitHub repository},
  url          = {https://github.com/imediacorp/FaCC}
}

---

Project Independence

This cosmological research work is completely independent and separate from any other projects. It originated as a simple thought experiment and hypothesis exploring whether the Golden Ratio might be fundamental to cosmic structure, motivated by self-similarity patterns observed from plants to galaxies. See INDEPENDENCE.md for a detailed statement of project independence.

---

Support

Questions or issues: please open a GitHub issue or email bryan@imediacorp.com.

---

Interactive Notebook

Run all tests in Google Colab (no setup needed):