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Curvature Relaxation Model (CRM)

A Four-Paper Program for Geometric Cosmology Without the Dark Sector

DOI License: CC BY 4.0

Overview

The Curvature Relaxation Model (CRM) replaces dark energy with a geometric curvature return potential and embeds modified gravity within the Horndeski/f(R) framework. The model is validated against Planck 2018 CMB data (TT+TE+EE, 6,405 data points) using hi_class with a custom crm_fR gravity model.

Key result: The native crm_fR model achieves Delta chi2 = -3.7 vs. LCDM on Planck CMB data (MCMC best-fit), with alpha_M_0 = 0.0011 +/- 0.0007 (1.76 sigma detection, P(alpha_M_0 > 0) = 100%) and 100*theta_s = 1.04173 (identical to LCDM).

Core Papers (I--IV)

Paper EN DE Topic
I papers/Paper1_EN.tex papers/Paper1_DE.tex Game-theoretic foundation, CRM, Pantheon+ validation
II papers/Paper2_EN.tex papers/Paper2_DE.tex MOND unification, baryon-only universe, running coupling
III papers/Paper3_EN.tex papers/Paper3_DE.tex Lagrangian (R + gamma R^2), scalaron dynamics, predictions
IV papers/Paper4_EN.tex papers/Paper4_DE.tex Galactic MOND from curvature saturation (DRAFT)

Extensions

Paper EN DE Topic DOI
V papers/extensions/Paper5_EN.tex papers/extensions/Paper5_DE.tex The Saturation Theorem: conditional projective-collar normal form for tanh saturation 10.5281/zenodo.19036188
VI papers/extensions/Paper6_EN.tex papers/extensions/Paper6_DE.tex QG-CRM: Ultraviolet Completion via Quantum Quadratic Gravity (DRAFT) 10.5281/zenodo.19352448

Paper V -- The Saturation Theorem identifies a conditional normal form for saturation dynamics. Axioms A--D, together with the signed interior composition assumptions, force an Abel/collar structure; the exact tanh representative follows once the additional projective boundary quotient D' is imposed. Major QG programs motivate the macroscopic A--D structure, while D' remains an explicit diagnostic/open physical ingredient rather than a derived microscopic theorem.

Paper VI -- QG-CRM: Ultraviolet Completion answers the open question from Paper V: "which UV completion selects k and Phi_0?" By identifying the gamma*R^2 sector of the CRM Lagrangian with asymptotically free quantum quadratic gravity (QQG), inflation is generated dynamically via RG running without an inflaton field. The Saturation Theorem provides the unique UV-IR interface. Predictions: n_s ~ 1 - 4/(3N) ~ 0.976, r >= 0.01, testable with Stage IV CMB experiments.

Key Results

Model chi2 (TT+TE+EE) Delta chi2 sigma8 100*theta_s
LCDM 6628.8 --- 0.811 1.04173
propto_omega cM=0.0002 6628.6 -0.2 0.826 1.04173
crm_fR n=0.5, aM0=0.001 6626.1 -2.7 0.899 1.04173
crm_fR n=1.0, aM0=0.0005 6627.1 -1.6 0.879 1.04173
crm_fR MCMC best-fit 6625.1 -3.7 --- 1.04173

The crm_fR model implements:

alpha_M(a) = alpha_M_0 * n * a^n / (1 + alpha_M_0 * a^n)
alpha_B(a) = -alpha_M(a) / 2     [f(R) relation]
alpha_T    = 0                     [c_gw = c, consistent with GW170817]
alpha_K    = 0

Installation

1. Python Dependencies

pip install -r requirements.txt

2. hi_class (Horndeski in CLASS Boltzmann code)

hi_class is required for CMB power spectrum computations and the crm_fR model.

# Clone hi_class
git clone https://github.com/miguelzuma/hi_class_public.git
cd hi_class_public

# Apply crm_fR patch (adds the native CRM gravity model)
python /path/to/crm-cosmology/scripts/patch_cfm.py

# Build hi_class with Python wrapper
cd python
python setup.py build

The patch modifies gravity_models_smg.c to add the crm_fR gravity model. See Patch Documentation below for details.

Tested with: hi_class v2.9.4+, Python 3.12, Cython 0.29.37, NumPy 1.26.4 on Ubuntu 24.04 (WSL).

3. Pantheon+ Data

The Pantheon+ supernova data (Scolnic et al. 2022) and Planck 2018 CMB spectra are downloaded automatically by the analysis scripts. No manual download required.

Downloaded external raw data are cached under data/raw/, which is intentionally ignored by Git. Versioned outputs remain in data/paper3/, results/, and figures/. For the SPARC Paper IV script, place the SPARC table and rotmod/ files under data/raw/sparc/ or set CRM_SPARC_DIR, CRM_SPARC_TABLE, or CRM_SPARC_ROTMOD_DIR.

Reproducing the Results

Paper I: CMB and MCMC

python scripts/paper1/run_full_mcmc.py            # Full MCMC (5 params, ~8h runtime)
python scripts/paper1/analyze_mcmc_results.py     # MCMC posterior analysis
python scripts/paper1/compute_TT_TE_EE.py         # Planck TT+TE+EE chi2 computation
python scripts/paper1/compute_fsigma8.py          # Growth rate f*sigma8
python scripts/paper1/full_cl_comparison.py        # Full Cl comparison cfm_fR vs LCDM

Paper II: Model Comparison

python scripts/paper2/compare_models.py            # LCDM vs constant_alphas vs cfm_fR
python scripts/paper2/plot_contour.py              # 2D chi2 contour from grid scan
python scripts/paper2/plot_tradeoff.py             # chi2-sigma8 tradeoff + convergence

Paper III: Pantheon+ and MOND

python scripts/paper3/cfm_pantheonplus_test.py     # CFM vs LCDM against Pantheon+ data
python scripts/paper3/cfm_baryon_only_test.py      # Baryon-only universe test
python scripts/paper3/cfm_mond_mcmc.py             # MCMC for CFM+MOND extended model
python scripts/paper3/scalaron_alphaM_theta_s.py   # theta_s resolution analysis
python scripts/paper3/poeschl_teller_path_integral.py  # sqrt(pi) path integral

Paper IV: Galactic MOND from Vector Sector

python scripts/paper4/sparc_full_analysis.py       # Full SPARC (171 galaxies) RAR test
python scripts/paper4/multi_galaxy_bvp.py          # Multi-mass BVP MOND attractor scan
python scripts/paper4/rotation_curves_bessel.py    # Bessel rotation curves
python scripts/paper4/a0_discrepancy.py            # a0 = cH0/(2pi) discrepancy analysis
python scripts/paper4/cfm_deep_mond_derivation.py  # Deep-MOND fixed point + Tully-Fisher

Infrastructure (cross-paper)

python scripts/patch_cfm.py                        # hi_class crm_fR gravity model patch
python scripts/test_cfm_fR_native.py               # Native crm_fR model test

Repository Structure

crm-cosmology/
  README.md                    # This file
  LICENSE                      # CC BY 4.0
  requirements.txt             # Python dependencies
  papers/                      # Core Papers I-IV (LaTeX + PDF, EN + DE)
    extensions/                # Extension Papers V-VI (and future)
  scripts/                     # Cross-paper infrastructure (patch, tests)
    paper1/                    # Paper I: CMB, MCMC analysis
    paper2/                    # Paper II: model comparison, plots
    paper3/                    # Paper III: Pantheon+, MOND, scalaron
    paper4/                    # Paper IV: galactic MOND, SPARC
  results/                     # Cross-paper results
    paper1/                    # Paper I: MCMC summaries, chi2 results
    paper3/                    # Paper III: baryon-only, MOND posteriors
    paper4/                    # Paper IV: SPARC, BVP, rotation curves
  figures/                     # Plots referenced in papers
    paper1/                    # Paper I: Cl spectra, fsigma8
    paper2/                    # Paper II: contours, tradeoffs
    paper3/                    # Paper III: MOND posteriors
  data/                        # Analysis outputs
    paper3/                    # Paper III: Pantheon+ fits

crm_fR Patch Documentation

The file scripts/patch_cfm.py applies 5 modifications to hi_class:

# File Location Change
0 include/background.h gravity_model enum Adds cfm_fR to the gravity model enum
1 gravity_models_smg.c gravity_models_init() Registers cfm_fR as a new gravity model (3 parameters, M2 evolution)
2 gravity_models_smg.c gravity_functions_smg() Computes alpha_M, alpha_B from parameters (alpha_M_0, n_exp, M*2_init)
3 gravity_models_smg.c gravity_print_stdout_smg() Adds print output for cfm_fR parameters
4 gravity_models_smg.c Error message Adds cfm_fR to the list of recognized models

Parameters passed via hi_class:

cosmo.set({
    'gravity_model': 'crm_fR',
    'parameters_smg': f'{alpha_M_0}, {n_exp}, 1.0',
    'expansion_model': 'lcdm',
    'Omega_smg': -1,
})

Physical interpretation:

  • alpha_M_0: Amplitude of the Planck mass running rate
  • n_exp: Power-law index controlling time evolution (n=0.5 best fit, n=1 reproduces propto_scale)
  • At early times (a << 1): alpha_M ~ alpha_M_0 * n * a^n (perturbative)
  • At late times (a ~ 1): alpha_M -> n_exp / (1 + alpha_M_0) (saturates)

Software Citations

This work uses the following open-source software:

  • CLASS (Cosmic Linear Anisotropy Solving System): Blas, Lesgourgues & Tram (2011), JCAP 07, 034. arXiv:1104.2933
  • hi_class (Horndeski in CLASS): Zumalacarregui, Bellini, Sawicki, Lesgourgues & Ferreira (2017), JCAP 01, 019. arXiv:1605.06102
  • emcee (MCMC sampler): Foreman-Mackey, Hogg, Lang & Goodman (2013), PASP 125, 306. arXiv:1202.3665
  • NumPy: Harris et al. (2020), Nature 585, 357.
  • SciPy: Virtanen et al. (2020), Nature Methods 17, 261.
  • Matplotlib: Hunter (2007), Computing in Science & Engineering 9, 90.

Observational data:

Citation

If you use this work, please cite the Zenodo deposit:

@misc{Geiger2026CRM,
  author    = {Geiger, Lukas},
  title     = {The Curvature Relaxation Model: A Four-Paper Program
               for Geometric Cosmology Without the Dark Sector},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.18728935},
  url       = {https://doi.org/10.5281/zenodo.18728935}
}

Individual papers:

@article{Geiger2026CRM_I,
  author  = {Geiger, Lukas},
  title   = {Game-Theoretic Cosmology and the Curvature Relaxation Model},
  year    = {2026},
  doi     = {10.5281/zenodo.18728935},
  note    = {Paper I of the CRM program}
}

@article{Geiger2026CRM_II,
  author  = {Geiger, Lukas},
  title   = {CRM-MOND Unification: A Baryonic Universe Without Dark Matter},
  year    = {2026},
  doi     = {10.5281/zenodo.18728935},
  note    = {Paper II of the CRM program}
}

@article{Geiger2026CRM_III,
  author  = {Geiger, Lukas},
  title   = {From Curvature Relaxation to Quantum Gravity: Lagrangian Foundations
             and Testable Predictions},
  year    = {2026},
  doi     = {10.5281/zenodo.18728935},
  note    = {Paper III of the CRM program}
}

@article{Geiger2026CRM_IV,
  author  = {Geiger, Lukas},
  title   = {The Galactic-Cosmological Nexus: Deriving MOND Dynamics
             from Curvature Saturation},
  year    = {2026},
  doi     = {10.5281/zenodo.18728935},
  note    = {Paper IV of the CRM program (draft)}
}

License

This work is licensed under CC BY 4.0.

Haftung / Liability

Dieses Projekt ist eine unentgeltliche Open-Source-Schenkung im Sinne der §§ 516 ff. BGB. Die Haftung des Urhebers ist gemäß § 521 BGB auf Vorsatz und grobe Fahrlässigkeit beschränkt. Ergänzend gelten die Haftungsausschlüsse aus GPL-3.0 / MIT / Apache-2.0 §§ 15–16 (je nach gewählter Lizenz).

Nutzung auf eigenes Risiko. Keine Wartungszusage, keine Verfügbarkeitsgarantie, keine Gewähr für Fehlerfreiheit oder Eignung für einen bestimmten Zweck.

This project is an unpaid open-source donation. Liability is limited to intent and gross negligence (§ 521 German Civil Code). Use at your own risk. No warranty, no maintenance guarantee, no fitness-for-purpose assumed.

About

Curvature Relaxation Model (CRM) -- Geometric alternative to dark energy and dark matter. Derives MOND (a0 = cH0/2pi) from f(R) gravity without free parameters. Validated against Planck CMB, Pantheon+ SN, and SPARC rotation curves.

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python research physics cosmology mcmc mond dark-energy

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