A geometric cosmological model derived from rₛ = 2R

Sean P. Myers | Independent Researcher | ORCID: 0009-0000-3132-5383

This repository contains the code for the paper:

"Hypersphere Cosmology: The Universe as the Three-Dimensional Hyperspherical Surface of a Four-Dimensional Hyperball" Myers, S.P. (2026). Submitted to JCAP. v14.5

The model derives a comprehensive cosmology from a single postulate: the Schwarzschild radius equals twice the cosmic radius at all scales (rₛ = 2R).

Note: D_C(z) = (c/H₀)·ln(1+z) is the unique comoving distance consistent with H(z) = H₀(1+z), derived from ∫c/H(z')dz'. An earlier version of the code used a phenomenological DM(z) = L·zc·log(1+z/zc) form that was inconsistent with the stated Hubble relation; this has been corrected.

• H₀ = 62.3 ± 3.2 km/s/Mpc (cosmic chronometers)
• Hubble tension resolved geometrically via γ(z) = √(2z+1)
• No dark matter or dark energy — replaced by gravitational field energy
• CMB acoustic scale consistent with Planck 2018 θ_MC (compressed likelihood)
• Low-ℓ CMB suppression predicted from boundary conditions on S³

├── run_hypersphere_fit.py       # CC + SNe Ia + BAO fitter (MAP + MCMC)
├── boundary_cmb.py              # Boundary Boltzmann CMB likelihood
├── next_suite_cmb.py            # Extended multi-model comparison suite
├── data/
│   ├── firas_monopole_spec_v1.txt   # COBE/FIRAS CMB spectrum
│   └── bao_model_and_data.csv       # Compiled BAO measurements
└── requirements.txt

What it does:

• Fits H₀ (and BAO sound horizon r_d) to cosmic chronometers, Pantheon+ SNe Ia, and BAO
• Uses D_C(z) = (c/H₀)·ln(1+z) — consistent with H(z) = H₀(1+z)
• Optimizer: scipy differential_evolution (global MAP) + Metropolis-Hastings MCMC for posteriors
• Reports χ²/DOF for CC, SN, BAO separately and combined

What it does NOT do:

• Does not use random search (older versions did — now replaced with real MCMC)
• Does not fit CMB power spectra (use boundary_cmb.py for CMB)

python run_hypersphere_fit.py --data-root ./data [--mcmc-steps 3000]

Outputs: hypersphere_fit_results.json, hypersphere_mcmc_chain.npy, hypersphere_mcmc_chain.csv

What it does:

• Implements the boundary radiation interpretation of the CMB
• Interprets the last-scattering surface as the S³/B⁴ boundary
• Computes the sound horizon r_d with γ(z) = √(2z+1) time dilation correction
• Fits to Planck 2018 compressed likelihood (θ_MC, ω_b, ω_cdm, A_s, n_s, τ)
• Computes TT power spectrum at low-ℓ (ℓ < 30) where boundary interpretation is physically motivated
• Provides an approximate template for ℓ = 30–200

Honest limitations (calibrated against CAMB):

• ✓ ℓ < 30 (Sachs-Wolfe plateau): boundary suppression model is physically motivated
• ✓ ℓ = 30–200 (first peak): acoustic scale θ_MC correctly computed
• ✗ ℓ > 200: power is off by 5–10× without full Boltzmann treatment
• ✗ Polarization at ℓ > 50 not reliably computed
• The χ²/DOF ≈ 1.6 reported in the paper applies to the compressed likelihood + low-ℓ evaluation only

python boundary_cmb.py --data-root ./data [--H0 62.3] [--plot]

Outputs: boundary_cmb_results.json, optionally boundary_cmb_spectrum.png

What it does:

• Compares ΛCDM against several phenomenological HEA distance models
• Uses compressed Planck likelihood, Pantheon+, DESI BAO
• Computes BAO residuals, distance duality tables, AIC/BIC
• Uses random search + optional Nelder-Mead refinement (not MCMC)

python next_suite_cmb.py --data-root ./data [--refine]

The fitting scripts require these data files (not included due to size):

Cosmic chronometer data is built in (Moresco 2023 compilation, 33 measurements).

Data sources:

• Pantheon+: Brout et al. 2022, https://github.com/PantheonPlusSH0ES/DataRelease
• DESI BAO DR1: DESI Collaboration 2024, https://data.desi.lbl.gov
• Planck 2018: Planck Collaboration 2020, A&A 641 A6

git clone https://github.com/Hypersphere-Cosmology/hypersphere-cosmology
cd hypersphere-cosmology
pip install -r requirements.txt
mkdir -p data

# Run CC-only fit (built-in data, no downloads needed)
python run_hypersphere_fit.py --data-root ./data --no-mcmc

# Run CMB compressed likelihood (built-in Planck 2018 values)
python boundary_cmb.py --data-root ./data

Reviewer Canvas123 (2026) raised three valid concerns addressed in this version:

1. DM(z)/H(z) inconsistency [FIXED]: run_hypersphere_fit.py now uses D_C(z) = (c/H₀)·ln(1+z), the analytic form consistent with H(z) = H₀(1+z).

2. Fake MCMC [FIXED]: run_hypersphere_fit.py now implements genuine Metropolis-Hastings MCMC with adaptive step sizes. MAP is found via scipy.differential_evolution.

3. Missing CMB code [ADDED]: boundary_cmb.py is the boundary Boltzmann implementation. It covers the compressed likelihood and low-ℓ TT spectrum. Full ℓ > 200 Boltzmann computation requires CAMB modifications (see limitations).

@article{Myers2026Hypersphere,
  title={Hypersphere Cosmology: The Universe as the Three-Dimensional 
         Hyperspherical Surface of a Four-Dimensional Hyperball},
  author={Myers, Sean P.},
  year={2026},
  doi={10.5281/zenodo.19322656},
  note={Submitted to JCAP},
  url={https://github.com/Hypersphere-Cosmology/hypersphere-cosmology}
}

MIT License.

Sean P. Myers | seanpmyers1975@gmail.com | ORCID: 0009-0000-3132-5383