CMB Anisotropies — Early Quintessence MCMC Inference

Author: Cherian Parangot Ittyipe Affiliation: Cardiff University, MSc Astrophysics (Merit, 2024)

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Overview

This repository implements a full Bayesian MCMC inference pipeline for constraining Early Quintessence (EDE) cosmological parameters using the Planck 2018 CMB temperature and polarisation power spectra.

It accompanies the MSc thesis:

Constraining Early Dark Energy with CMB Anisotropies
Cherian Parangot Ittyipe, Cardiff University, 2024
DOI: 10.5281/zenodo.17681763

Key results (from thesis)

Parameter	Posterior mean ± 1σ
$H_0$	69.56 ± 0.93 km/s/Mpc
$f_\mathrm{EDE}(z_c)$	0.083 ± 0.024
$\log_{10} z_c$	3.47 ± 0.10
$n$	2.95 ± 0.24

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Physics

Early Quintessence is a class of dark energy models where a scalar field with a tracking equation of state contributes a non-negligible fraction of the total energy density around the epoch of matter–radiation equality ($z \sim 3500$). This injected energy modifies the sound horizon and shifts the CMB acoustic peaks, offering a potential resolution of the Hubble tension.

The model is parameterised by:

• $f_\mathrm{EDE}(z_c)$: fraction of dark energy at critical redshift $z_c$
• $\log_{10} z_c$: location of the EDE peak
• $n$: power-law index of the scalar potential $V(\phi) \propto [1 - \cos(\phi/f)]^n$
• $\theta_i = \phi_i/f$: initial field displacement

Theory is computed using CAMB with the built-in EarlyQuintessence dark energy module (Smith et al. 2020).

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Repository Structure

cmb_ede_mcmc/
├── mcmc/
│   ├── __init__.py
│   ├── theory.py        # CAMB interface (EarlyQuintessence D_ell)
│   ├── data.py          # Planck 2018 FITS loader
│   ├── likelihood.py    # log-prior, log-likelihood, posterior factory
│   └── sampler.py       # emcee runner + chain saving utilities
├── analysis/
│   ├── __init__.py
│   ├── diagnostics.py   # R-hat, ESS, posterior summary
│   └── plotting.py      # traces, corner, spectrum plots
├── notebooks/
│   ├── 01_run_mcmc.ipynb    # interactive MCMC run
│   └── 02_analysis.ipynb    # diagnostics, corner plot, spectrum
├── data/                    # (gitignored) local Planck FITS data
├── results/                 # (gitignored) chain outputs
├── run_mcmc.py              # CLI entry point
├── params.json              # thesis best-fit parameter set (fixed)
├── plot_cmb_from_posterior.py  # standalone spectrum plotter
├── requirements.txt
└── README.md

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Installation

1. Clone the repository

git clone https://github.com/Cherian-pi/cmb-ede-mcmc.git
cd cmb-ede-mcmc

2. Create a virtual environment (recommended)

python -m venv .venv
source .venv/bin/activate        # Linux/macOS
.venv\Scripts\activate           # Windows

3. Install dependencies

pip install -r requirements.txt

Note: CAMB is included in requirements.txt and installs the EarlyQuintessence
module automatically on Python ≥ 3.10.

4. Obtain Planck 2018 data

Download COM_PowerSpect_CMB_R2.02.fits from the Planck Legacy Archive (free registration required).
Place it anywhere and set the path as described below.

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Quick Start

Option A: Command line

# Set path to Planck FITS file
export PLANCK_FITS=/path/to/COM_PowerSpect_CMB_R2.02.fits

# Full run (TT + TE + EE, 36 walkers, 500 steps)
python run_mcmc.py

# Quick functional test (not converged)
python run_mcmc.py --burnin 10 --nsteps 50 --nwalkers 22

# Custom output directory
python run_mcmc.py --outdir results/run_TT_only --spectra TT

See python run_mcmc.py --help for all options.

Option B: Jupyter notebooks

jupyter lab
# Open notebooks/01_run_mcmc.ipynb

Edit the PLANCK_FITS path in cell 0, then run all cells.
After the chain completes, open notebooks/02_analysis.ipynb for diagnostics and plots.

Option C: Reproduce the thesis spectrum only (no MCMC)

python plot_cmb_from_posterior.py

This reads params.json (thesis best-fit parameters) and plots the CMB TT spectrum against Planck 2018 data without running any MCMC.

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Parameter Space

The inference runs over a 10-dimensional parameter vector:

#	Parameter	Description	Prior
0	$H_0$	Hubble constant [km/s/Mpc]	Flat [40, 90]
1	$\Omega_b h^2$	Baryon density	Flat [0.01, 0.04]
2	$\Omega_c h^2$	CDM density	Flat [0.05, 0.30]
3	$\tau$	Optical depth	Flat + Gauss(0.0544, 0.0073)
4	$n_s$	Spectral index	Flat [0.89, 1.10]
5	$A_s$	Primordial amplitude	Flat [1×10⁻⁹, 3×10⁻⁹]
6	$\log_{10} z_c$	EDE critical redshift	Flat [2.8, 3.8]
7	$f_\mathrm{EDE}(z_c)$	EDE fraction at $z_c$	Flat [0.0, 0.20]
8	$n$	EDE potential power index	Flat [1.5, 4.0]
9	$\theta_i$	Initial field displacement [rad]	Flat [0, π]

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Runtime

Each CAMB call takes approximately 0.5–2 seconds on a modern CPU.
A typical converged run (36 walkers × 500 production steps) requires ~9,000 CAMB evaluations and takes 3–10 hours on a single core.

For faster iteration:

• Use --nsteps 50 --burnin 10 for a quick functional test (~20 minutes)
• Use multiprocessing: emcee.EnsembleSampler(pool=multiprocessing.Pool())
• Run on HPC with MPI via schwimmbad.MPIPool

Downstream

The best-fit parameter set (params.json) from this inference is used to generate the CMB TT power spectrum in the companion repository: → github.com/Cherian-pi/cmb-spectrum

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Output Files

After a run, results/ contains:

File	Description
EDE_Planck_<ts>_flat.csv	Flat chain (n_samples × 11): parameters + log_posterior
EDE_Planck_<ts>_chain.npy	Raw chain (nsteps, nwalkers, ndim)
EDE_Planck_<ts>_meta.json	Run metadata (settings, acceptance fraction)
traces.png	Walker trace plots
corner.png	2D marginal posteriors
spectrum.png	Best-fit CMB spectrum vs Planck data

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References

• Planck 2018: Aghanim et al. 2020, A&A 641 A5 (arXiv:1807.06209)
• Early Quintessence: Smith, Poulin & Amin 2020, PRD 101 063523 (arXiv:1908.06995)
• CAMB: Lewis, Challinor & Lasenby 2000; Howlett et al. 2012 (camb.info)
• emcee: Foreman-Mackey et al. 2013 (arXiv:1202.3665)
• Thesis: Ittyipe 2024, DOI:10.5281/zenodo.17681763

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License

MIT — see LICENSE.

cmb-ede-mcmc