Bayesian MCMC for cosmology

Monte Carlo methods for Bayesian inference: a from-scratch Metropolis–Hastings test of dark energy evolution.

A hand-written Metropolis–Hastings sampler is used to fit the cosmic expansion rate $H(z)$ and to ask a concrete question: do the data require dynamical dark energy, or is a cosmological constant enough? The cosmology is the application; the focus is correct Bayesian inference and rigorous MCMC diagnostics.

Results

Fitting ΛCDM to 36 cosmic-chronometer + BAO $H(z)$ measurements (Yu, Ratra & Wang 2018):

Parameter	Posterior (mean ± 1σ)
$H_0$	71.9 ± 1.4 km s⁻¹ Mpc⁻¹
$\Omega_m$	0.229 ± 0.010
$\chi^2/\mathrm{dof}$	0.68

Sampling used 16 chains; all parameters converged (Gelman–Rubin $\hat{R} < 1.02$).

Does the data justify dynamical dark energy?

Comparing ΛCDM against the CPL ($w_0$–$w_a$) dynamical dark-energy model:

Model	k	χ²/dof	AIC	BIC	ΔBIC
ΛCDM	2	0.68	26.98	30.15	0.00
CPL	4	0.63	28.11	34.44	+4.29

CPL lowers $\chi^2$ only marginally — not enough to pay for its two extra parameters. ΔBIC ≈ +4.3 favours ΛCDM: the data do not justify dynamical dark energy. With this small $H(z)$ sample the CPL equation-of-state parameters $(w_0, w_a)$ are essentially unconstrained (a strongly degenerate posterior), consistent with the need for combined probes (BAO + CMB + SNe) to break the degeneracy.

Method

• Data — 36 cosmic-chronometer and BAO $H(z)$ points, data/hubble_cosmic_chronometers.txt.
• Sampler — random-walk Metropolis–Hastings written from scratch (src/mcmc/sampler.py), in log-likelihood space for numerical stability, with burn-in, acceptance tracking, and uniform-prior bounds. CPL's degenerate posterior is sampled with a preconditioned proposal (covariance estimated from a pilot run).
• Convergence — many-chain Gelman–Rubin $\hat{R}$ (with running-$\hat{R}$ and trace diagnostics).
• Model selection — best-fit $\chi^2$, AIC and BIC (src/mcmc/inference.py).

Repository layout

src/mcmc/                 # reusable package
  ├── models.py           # ΛCDM, CPL (w0-wa) H(z) models
  ├── inference.py        # chi², likelihood, best-fit, AIC/BIC
  ├── sampler.py          # Metropolis–Hastings, multi-chain, Gelman–Rubin
  └── plotting.py         # traces, R-hat, H(z) band, credible-region marginals
data/                     # H(z) measurements
figures/                  # generated figures
tests/                    # unit tests for the sampler and inference
main.ipynb   # narrated end-to-end demonstration

Usage

pip install -r requirements.txt
python -m pytest tests/          # run the unit tests
jupyter notebook main.ipynb

Reference

Yu, H., Ratra, B., & Wang, F. Y. (2018). Hubble parameter and Baryon Acoustic Oscillation measurement constraints on the Hubble constant, the deviation from the spatially flat ΛCDM model, the deceleration–acceleration transition redshift, and spatial curvature. The Astrophysical Journal, 856(1), 3.