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).
| Relation | Formula |
|---|---|
| Hubble parameter | H(z) = H₀(1+z) |
| Comoving distance | D_C(z) = (c/H₀) · ln(1+z) |
| Hubble distance | D_H(z) = c / H(z) = c / (H₀(1+z)) |
| Luminosity distance | D_L(z) = (1+z) · D_C(z) |
| Time dilation | γ(z) = √(2z+1) |
| H₀ (CC fit) | 62.3 ± 3.2 km/s/Mpc |
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.pyfor 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):
| Script | Required files |
|---|---|
run_hypersphere_fit.py |
data/pantheon_plus/Pantheon+SH0ES.dat, data/pantheon_plus/Pantheon+SH0ES_STAT+SYS.cov, data/desi_bao/desi_bao_summary.csv |
boundary_cmb.py |
data/planck_acoustic/planck_compressed.csv (optional; built-in Planck 2018 values used if absent) |
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| Prediction | Test | Timeline |
|---|---|---|
| H(z) = H₀(1+z) | DESI DR3 Hubble diagram | ~2027 |
| r = 0 (no primordial B-modes) | CMB-S4 | ~2030 |
| Low-ℓ TT suppression (ℓ=2,3) | Future full-sky CMB | ~2028+ |
| 22% directional H₀ variation | DESI + Euclid | ~2028 |
| 3.3% smaller GW distances at z~1 | LIGO O5 / Einstein Telescope | ~2027–2035 |
Reviewer Canvas123 (2026) raised three valid concerns addressed in this version:
-
DM(z)/H(z) inconsistency [FIXED]:
run_hypersphere_fit.pynow usesD_C(z) = (c/H₀)·ln(1+z), the analytic form consistent with H(z) = H₀(1+z). -
Fake MCMC [FIXED]:
run_hypersphere_fit.pynow implements genuine Metropolis-Hastings MCMC with adaptive step sizes. MAP is found viascipy.differential_evolution. -
Missing CMB code [ADDED]:
boundary_cmb.pyis 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