Research code and papers for a geometric modified-gravity cosmology project
The Curvature Relaxation Model (CRM) is a research program in geometric cosmology and modified gravity. It investigates whether parts of dark-energy and dark-matter phenomenology can be modeled through curvature relaxation, scalaron dynamics, and a MOND-oriented vector sector rather than by introducing separate dark-sector components.
This repository contains the bilingual paper series, the crm_fR / cfm_fR analysis scripts, result tables, and figures for CMB, Pantheon+, MOND, and SPARC-related checks.
Current headline CMB result in this repository: the native crm_fR model yields Delta chi2 = -3.7 relative to LCDM on Planck 2018 CMB TT+TE+EE data in the current MCMC best-fit run, with alpha_M_0 = 0.0011 +/- 0.0007 and 100*theta_s = 1.04173.
Research status: this is an open research/preprint repository, not a consensus cosmology package. Theoretical and statistical limitations are summarized in the paper texts and public result artifacts included here; additional working notes remain local. The live Zenodo v7 record has not yet been updated to the latest local paper rebuilds.
- Modified-gravity reproduction: scripts and figures for Planck 2018 CMB, Pantheon+ supernova, MOND, and SPARC-oriented analyses.
- Paper archive: English and German LaTeX/PDF papers for the CRM program and extension papers.
- hi_class patching: a patch workflow for adding the
crm_fRmodel to hi_class. - Review trail: project notes, publication strategy, and open review items for follow-up work.
| CMB spectrum comparison | MCMC posterior |
|---|---|
 |
 |
| MOND posterior | SPARC RAR comparison |
|---|---|
 |
 |
| 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) |
| 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, imply an Abel/collar structure; the exact tanh representative appears once the additional projective boundary quotient D' is imposed. Major QG programs motivate the macroscopic A--D structure, while D' remains an explicit diagnostic and open physical ingredient rather than a derived microscopic theorem.
Paper VI -- QG-CRM: Ultraviolet Completion addresses an open question from Paper V: which UV completion selects k and Phi_0? It explores a proposed identification of the gamma*R^2 sector of the CRM Lagrangian with asymptotically free quantum quadratic gravity (QQG), under which inflation is generated dynamically via RG running without an inflaton field. In this draft, the Saturation Theorem is treated as the UV-IR interface. The resulting headline predictions are n_s ~ 1 - 4/(3N) ~ 0.976 and r >= 0.01, testable with Stage IV CMB experiments.
| 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
pip install -r requirements.txthi_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 buildThe 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).
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.
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 LCDMpython 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 + convergencepython 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 integralpython 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-Fisherpython scripts/patch_cfm.py # hi_class crm_fR gravity model patch
python scripts/test_cfm_fR_native.py # Native crm_fR model testcrm-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
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 raten_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)
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:
- Pantheon+: Scolnic et al. (2022), ApJ 938, 113. arXiv:2112.03863
- Planck 2018: Aghanim et al. (2020), A&A 641, A6. arXiv:1807.06209
- SPARC: Lelli, McGaugh & Schombert (2016), AJ 152, 157. arXiv:1606.09251
If you use this work, please cite the Zenodo deposit and include the accessed Git commit when referring to the repository code. GitHub can also read the repository-level CITATION.cff file for citation export.
- Concept DOI for all CRM I--IV versions: 10.5281/zenodo.18728935
- Latest published Zenodo v7.0 record checked for this README: 10.5281/zenodo.19233559
@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},
version = {7.0},
doi = {10.5281/zenodo.19233559},
url = {https://doi.org/10.5281/zenodo.19233559}
}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)}
}This work is licensed under CC BY 4.0.
Dieses Projekt ist eine unentgeltliche Open-Science-Veröffentlichung. Die Haftung des Urhebers ist gemäß § 521 BGB auf Vorsatz und grobe Fahrlässigkeit beschränkt.
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.