🌌 Xi Dark Energy BAO MCMC

Exploratory Bayesian comparison of ΛCDM, CPL and an oscillating dark-energy equation of state

using Pantheon+SH0ES supernovae and DESI DR2 BAO

A compact research-grade pipeline for testing whether an oscillating dark-energy equation of state can compete with standard cosmological parameterizations when confronted with modern distance probes.

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1. Overview

This repository implements an exploratory cosmological analysis combining:

• Pantheon+SH0ES Type Ia supernovae, using distance moduli and the full statistical + systematic covariance matrix;
• DESI DR2 BAO measurements, using D_V/r_d, D_M/r_d, and D_H/r_d
• Bayesian MCMC inference with emcee;
• Model comparison using χ², AIC and BIC;
• Convergence diagnostics based on acceptance fractions and integrated autocorrelation times.

The goal is to compare three models:

Model	Description	Free parameters
ΛCDM	Standard flat cosmology with w = −1	2
CPL	Dynamical dark energy with w(z) = w₀ + wₐ z/(1+z)	4
Ξosc	Phenomenological oscillating dark-energy equation of state	6

The oscillating model is written as:

$$ w(t)=w_{0,\mathrm{mean}}+\epsilon\cos\left(\frac{2\pi t}{T}+\phi_0\right) $$

where ε controls the oscillation amplitude, T the characteristic period and φ₀ the phase.

Important

This project does not claim a discovery of oscillating dark energy.
It is an exploratory, reproducible test of whether a phenomenological oscillating equation of state remains competitive against standard dark-energy parameterizations.

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2. Scientific motivation

The standard cosmological model, ΛCDM, assumes that the accelerated expansion of the Universe is driven by a cosmological constant with:

$$ w=-1 $$

However, current and upcoming distance probes motivate careful tests of whether the dark-energy equation of state could evolve with cosmic time.

A common phenomenological extension is the Chevallier–Polarski–Linder (CPL) parameterization:

$$ w(z)=w_0+w_a\frac{z}{1+z} $$

This project adds a third possibility:

$$ w(t)=w_{0,\mathrm{mean}}+\epsilon\cos\left(\frac{2\pi t}{T}+\phi_0\right) $$

which asks a simple question:

Could current supernova + BAO distance data accommodate a small oscillatory component in the dark-energy equation of state?

The analysis is intentionally modest and transparent: it does not try to establish a full physical theory, but rather tests a phenomenological signature against observational data.

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3. Data used

Pantheon+SH0ES supernovae

The supernova component uses the Pantheon+SH0ES distance-modulus catalogue. The script reads:

• zHD
• MU_SH0ES or MU_SHOES
• MU_SH0ES_ERR_DIAG or MU_SHOES_ERR_DIAG

and applies the redshift cut:

$$ z>0.01 $$

For the final run:

$$ N_{\rm SNe}=1590 $$

The full statistical + systematic covariance matrix is used.

DESI DR2 BAO

The BAO component uses 13 DESI DR2 BAO measurements of the form:

• D_V/r_d
• D_M/r_d
• D_H/r_d

with a fixed fiducial sound horizon:

$$ r_d = 147.09\ \mathrm{Mpc} $$

For the final combined analysis:

$$ N_{\rm total}=1590+13=1603 $$

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4. Method

For each model, the pipeline:

1. defines the expansion history E(z)=H(z)/H₀;
2. computes cosmological distances in a flat universe;
3. predicts supernova distance moduli (μ(z));
4. predicts BAO observables D_V/r_d, D_M/r_d, and D_H/r_d;
5. evaluates the combined Gaussian likelihood using covariance matrices;
6. samples the posterior using MCMC;
7. summarizes marginalized constraints;
8. compares models using χ², AIC and BIC.

The model-comparison criteria are:

$$ \mathrm{AIC}=\chi^2+2k $$

and

$$ \mathrm{BIC}=\chi^2+k\ln N $$

where k is the number of fitted parameters and N is the total number of data points.

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5. Final exploratory run

The final long run used:

python src/xi_protocol1b_bao_run.py --nsteps 100000 --burn-in 20000 --nwalkers 96

with:

Quantity	Value
Supernovae retained	1590
BAO measurements	13
Total data points	1603
MCMC steps	100000
Burn-in	20000
Walkers	96
Samples retained for Ξosc	7680000

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6. Main results

Model comparison

Model	χ²	k	AIC	BIC	ΔAIC
CPL	1531.824	4	1539.824	1561.343	0.000
Ξosc	1529.763	6	1541.763	1574.041	1.939
ΛCDM	1554.165	2	1558.165	1568.924	18.341

Key interpretation:

• Ξ_osc gives the lowest raw χ².
• CPL remains slightly preferred by AIC.
• The difference between CPL and Ξ_osc is small:

$$ \Delta\mathrm{AIC}\approx1.94 $$

• BIC is more severe and favors CPL because Ξ_osc has two additional parameters.
• ΛCDM is disfavored relative to the two dynamical models in this exploratory setup.

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7. Oscillation-amplitude diagnostic

For the oscillating model, the final marginalized constraint is:

ε = 0.1847^+0.0768_−0.0867

Using a symmetrized uncertainty:

$$ \sigma_\epsilon \approx 0.0818 $$

which gives:

$$ \frac{\epsilon}{\sigma_\epsilon}\approx2.26 $$

This is an interesting non-zero preference, but it is not strong enough to claim a detection.

A conservative reading is:

The combined Pantheon+SH0ES + DESI DR2 BAO dataset allows an oscillatory dark-energy component and gives it the best raw χ², but the evidence is not decisive once model complexity is penalized.

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8. Figures

Posterior constraints for Ξ_osc

Reconstructed equation of state

DESI DR2 BAO residuals

Additional figures are available in:

figures/protocol1b_pantheon_desi/
figures/protocol1_pantheon_only/
figures/diagnostics/

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9. Repository structure

xi-dark-energy-bao-mcmc/
│
├── README.md
├── LICENSE
├── CITATION.cff
├── requirements.txt
├── environment.yml
├── pyproject.toml
├── Makefile
│
├── src/
│   ├── xi_protocol1b_bao_run.py
│   ├── xi_protocol1_run.py
│   ├── 00_load_pantheon.py
│   └── xi_cosmo/
│       ├── results.py
│       ├── diagnostics.py
│       ├── labels.py
│       └── plotting.py
│
├── scripts/
│   ├── check_environment.py
│   ├── download_data.py
│   ├── export_final_tables.py
│   ├── make_chain_diagnostics.py
│   ├── organize_existing_outputs.py
│   ├── run_quick.ps1
│   └── run_final.ps1
│
├── data/
│   ├── README.md
│   └── raw/
│
├── figures/
│   ├── protocol1_pantheon_only/
│   ├── protocol1b_pantheon_desi/
│   └── diagnostics/
│
├── results/
│   └── protocol1b_pantheon_desi/
│       ├── final_model_comparison.csv
│       ├── final_parameter_estimates.csv
│       ├── final_model_comparison.md
│       └── chains/
│
├── notebooks/
│   ├── 00_project_overview.ipynb
│   ├── 01_data_inspection.ipynb
│   ├── 02_results_review.ipynb
│   └── 03_mcmc_diagnostics.ipynb
│
├── reports/
│   ├── Protocole1b_Pantheon_DESI_DR2_AlexBaker.pdf
│   └── latex/
│
├── docs/
│
└── tests/

---

10. Installation

Option A — Standard Python environment

git clone https://github.com/PhD-Brown/xi-dark-energy-bao-mcmc.git
cd xi-dark-energy-bao-mcmc

python -m venv .venv

On Windows PowerShell:

.\.venv\Scripts\Activate.ps1

On macOS/Linux:

source .venv/bin/activate

Then install:

python -m pip install --upgrade pip
pip install -r requirements.txt
pip install -e .

Option B — Conda

conda env create -f environment.yml
conda activate xi-dark-energy
pip install -e .

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11. Data setup

The raw data files are not committed by default.

Required Pantheon+SH0ES files

Place the following files in:

data/raw/

Required files:

Pantheon+SH0ES.dat
Pantheon+SH0ES_STAT+SYS.cov

DESI DR2 BAO files

The DESI BAO files can be downloaded automatically:

python scripts/download_data.py

This creates:

data/raw/desi_gaussian_bao_ALL_GCcomb_mean.txt
data/raw/desi_gaussian_bao_ALL_GCcomb_cov.txt

See data/README.md for details.

---

12. Running the analysis

Quick smoke test

Use this to verify that the pipeline runs end-to-end without launching a long MCMC:

python src/xi_protocol1b_bao_run.py --quick

On Windows:

.\scripts\run_quick.ps1

Medium test

python src/xi_protocol1b_bao_run.py --nsteps 5000 --burn-in 1000 --nwalkers 64

Final long run

python src/xi_protocol1b_bao_run.py --nsteps 100000 --burn-in 20000 --nwalkers 96

On Windows:

.\scripts\run_final.ps1

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13. Outputs

The main runtime script writes outputs to:

results/
figures/

including:

results/summary_protocol1b_combined.txt
results/model_comparison_combined.csv
results/chains_<MODEL>_combined.npy

figures/corner_<MODEL>_combined.png
figures/w_of_z_combined.png
figures/bao_residuals_combined.png

Curated final outputs used for the report and public notebooks are organized under:

results/protocol1b_pantheon_desi/
figures/protocol1b_pantheon_desi/

Large MCMC chain files are intentionally ignored by Git:

results/**/chains/*.npy

They should be kept locally or distributed through a GitHub Release or external archive if needed.

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14. Notebooks

The repository includes four lightweight notebooks:

Notebook	Purpose
00_project_overview.ipynb	Public-facing overview of the final results
01_data_inspection.ipynb	Checks local data files and basic dimensions
02_results_review.ipynb	Reviews final tables and model-comparison results
03_mcmc_diagnostics.ipynb	Inspects saved MCMC chains if available locally

The overview notebook does not rerun the full MCMC. It loads final summary tables and reproduces the key interpretation.

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15. Diagnostics

The final long run gave for Ξ_osc:

acceptance fraction ≈ 0.180
tau_max ≈ 1504
N/50 = 2000

Thus the practical criterion:

$$ N/50 > \tau_{\max} $$

is satisfied for the final run.

However, some parameters remain weakly constrained, especially:

• T
• φ₀

This is expected for an exploratory oscillatory model with phase and period freedom.

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16. Testing

Run the local checks with:

python scripts/check_environment.py
python scripts/export_final_tables.py
python scripts/make_chain_diagnostics.py
python -m pytest tests -q
python -m compileall src scripts

The repository also includes a GitHub Actions workflow:

.github/workflows/python-check.yml

which installs the package and runs the test suite.

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17. Scientific caveats

This project should be read as a methodological and exploratory analysis, not as a definitive cosmological claim.

Important caveats:

1. ε ≈ 2.3σ is suggestive but not decisive.
2. T and φ₀ remain weakly constrained.
3. BIC favors CPL over Ξ_osc.
4. The fixed r_d choice should be varied in future work.
5. Additional probes should be included before drawing strong physical conclusions.
6. The Ξ_osc model is phenomenological and should eventually be connected to a more complete theoretical framework.

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18. Future work

Possible extensions include:

• varying or marginalizing over r_d;
• adding CMB priors;
• adding cosmic chronometers H(z);
• testing alternative BAO covariance treatments;
• comparing with more flexible dark-energy reconstructions;
• performing prior-sensitivity analyses on ε, T, and φ₀;
• using nested sampling for a stronger Bayesian evidence comparison;
• testing whether oscillatory behavior survives under alternative data combinations.

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19. Project status

This repository is currently at the level of:

v0.1.0 — Initial exploratory Pantheon+SH0ES + DESI DR2 BAO analysis

It is suitable as:

• a reproducible research mini-project;
• a portfolio project in observational cosmology;
• a demonstration of Bayesian inference with real astrophysical data;
• a starting point for more robust dark-energy model comparison.

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20. Citation

If you use or adapt this repository, please cite it using the metadata in:

CITATION.cff

A suggested citation format is:

Baker, A. (2026). Xi Dark Energy BAO MCMC: Exploratory Bayesian comparison of
ΛCDM, CPL and an oscillating dark-energy equation of state using Pantheon+SH0ES
and DESI DR2 BAO.

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21. License

This project is released under the MIT License. See LICENSE.

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22. Acknowledgements

This repository was developed as an exploratory cosmology analysis project using public distance-probe datasets and open-source scientific Python tools, including:

• numpy
• pandas
• scipy
• matplotlib
• astropy
• emcee
• corner

The analysis is intended as a transparent and reproducible starting point for further discussion, validation and extension.

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Scientific takeaway

Ξ_osc achieves the lowest raw χ², remains close to CPL according to AIC, and shows a non-zero oscillation amplitude at approximately 2.3σ.

However, CPL remains preferred by BIC, and the result should be interpreted as an exploratory indication rather than a detection.

Exploratory. Reproducible. Open to refinement.