[Paper Review] Markov Chain Monte Carlo methods applied to measuring the fine structure constant from quasar spectroscopy

Julian A. King, D. Mortlock+2 more|ArXiv.org|Oct 14, 2009

Galaxies: Formation, Evolution, Phenomena3 citations

TL;DR

This paper applies Markov Chain Monte Carlo (MCMC) methods to validate the reliability of VPFIT, a spectral fitting software used to measure cosmological variations in the fine structure constant α from quasar absorption lines. It confirms that VPFIT produces consistent and accurate estimates of Δα/α and its uncertainties, supporting the robustness of prior results suggesting a potential variation in α.

ABSTRACT

Recent attempts to constrain cosmological variation in the fine structure constant, alpha, using quasar absorption lines have yielded two statistical samples which initially appear to be inconsistent. One of these samples was subsequently demonstrated to not pass consistency tests; it appears that the optimisation algorithm used to fit the model to the spectra failed. Nevertheless, the results of the other hinge on the robustness of the spectral fitting program VPFIT, which has been tested through simulation but not through direct exploration of the likelihood function. We present the application of Markov Chain Monte Carlo (MCMC) methods to this problem, and demonstrate that VPFIT produces similar values and uncertainties for (Delta alpha)/(alpha), the fractional change in the fine structure constant, as our MCMC algorithm, and thus that VPFIT is reliable.

Motivation & Objective

• To assess the reliability of VPFIT, a widely used spectral fitting tool, in estimating Δα/α from quasar absorption lines.
• To test whether the optimisation algorithm in VPFIT fails to converge to the true maximum-likelihood solution, as previously suspected in some studies.
• To use MCMC methods as an independent validation of statistical uncertainties and parameter estimates produced by VPFIT.
• To determine whether the discrepancy between Keck and VLT measurements of Δα/α is due to algorithmic failure or physical variation.
• To confirm that VPFIT's reported uncertainties on Δα/α are statistically sound, even when other parameters (e.g., column densities) show non-Gaussian distributions.

Proposed method

• Employed a Metropolis-Hastings MCMC algorithm to sample the posterior distribution of model parameters, including Δα/α, for absorption systems in quasars.
• Used the likelihood function L(𝐱) ≡ exp(−χ²(𝐱)/2), where χ² is the standard reduced chi-squared statistic, to define the posterior distribution.
• Fitted multiple absorption lines (e.g., Si ii, Al iii, Fe ii, Mg i) across three quasars: LBQS 2206−1958, LBQS 0013−0029, and Q 0551−366, using Voigt profiles with up to three components per transition.
• Performed convergence diagnostics and examined the marginal posterior distributions of Δα/α and other parameters to assess Gaussianity and reliability.
• Compared MCMC-derived estimates of Δα/α and its uncertainties directly with those from VPFIT to validate consistency.
• Used weighted mean combination of MCMC results to estimate a combined Δα/α across systems, assuming a common α value.

Experimental results

Research questions

• RQ1 Does the VPFIT optimisation algorithm reliably converge to the true maximum-likelihood solution for Δα/α in quasar absorption line data?
• RQ2 Are the statistical uncertainties on Δα/α reported by VPFIT accurate, especially when other model parameters (e.g., column density) are non-Gaussian?
• RQ3 Can MCMC methods independently confirm the reliability of VPFIT’s parameter estimates in complex spectral fitting scenarios?
• RQ4 Is the discrepancy between Keck and VLT measurements of Δα/α due to algorithmic failure in VPFIT or a real cosmological variation?
• RQ5 To what extent do non-Gaussian parameter distributions affect the reliability of Δα/α estimates, and is Δα/α itself approximately Gaussian?

Key findings

• MCMC results for Δα/α in the three quasar absorption systems (LBQS 2206−1958, LBQS 0013−0029, Q 0551−366) are consistent with those from VPFIT, confirming VPFIT’s reliability.
• Despite significant non-Gaussianity in the posterior distributions of column densities and Doppler parameters, the marginal distribution of Δα/α remains approximately Gaussian.
• The MCMC method successfully identified and corrected for potential algorithmic failures in spectral fitting, validating VPFIT’s performance.
• The combined estimate of Δα/α from the three systems is (−0.74 ± 0.59) × 10⁻⁵, statistically consistent with no change in α.
• The study confirms that VPFIT’s reported uncertainties on Δα/α are trustworthy, even when other parameters are non-Gaussian or poorly constrained.
• The results support the conclusion that the earlier detection of Δα/α ≈ (−0.57 ± 0.11) × 10⁻⁵ by Murphy et al. (2004) is not due to failure of the VPFIT optimisation algorithm.