Mapping systematic errors in helium abundance determinations using Markov Chain Monte Carlo

TL;DR

This paper addresses the challenge of precisely determining the primordial helium abundance $Y_p$ from extragalactic H II regions by quantifying both statistical and systematic uncertainties. It introduces Markov Chain Monte Carlo (MCMC) to jointly sample eight physical and observational parameters, using a $\chi^2$-based likelihood that incorporates reddening, absorption, and radiative-transfer corrections, with $W(H\beta)$ treated as a nuisance parameter. The authors demonstrate that MCMC reveals degeneracies and potential secondary minima (especially at large optical depth $\tau$) and show that a conservative prior on $T_{\mathrm{OIII}}$ can remove unphysical minima without introducing significant bias. Applied to real observations, the method yields robust $y^+$ values and realistic uncertainty estimates, while still indicating sizable systematic uncertainties in $Y_p$ with current spectra. Overall, the MCMC framework provides a statistically rigorous path toward more reliable primordial helium measurements and can scale with higher-quality data to improve precision in cosmological tests of Big Bang Nucleosynthesis.

Abstract

Monte Carlo techniques have been used to evaluate the statistical and systematic uncertainties in the helium abundances derived from extragalactic H~II regions. The helium abundance is sensitive to several physical parameters associated with the H~II region. In this work, we introduce Markov Chain Monte Carlo (MCMC) methods to efficiently explore the parameter space and determine the helium abundance, the physical parameters, and the uncertainties derived from observations of metal poor nebulae. Experiments with synthetic data show that the MCMC method is superior to previous implementations (based on flux perturbation) in that it is not affected by biases due to non-physical parameter space. The MCMC analysis allows a detailed exploration of degeneracies, and, in particular, a false minimum that occurs at large values of optical depth in the He~I emission lines. We demonstrate that introducing the electron temperature derived from the [O~III] emission lines as a prior, in a very conservative manner, produces negligible bias and effectively eliminates the false minima occurring at large optical depth. We perform a frequentist analysis on data from several "high quality" systems. Likelihood plots illustrate degeneracies, asymmetries, and limits of the determination. In agreement with previous work, we find relatively large systematic errors, limiting the precision of the primordial helium abundance for currently available spectra.

Figures (11)

• Figure 1: $\chi^{2}$ versus abundance for synthetic data with model parameters, $y^{+}=0.08$, $n_{e} = 100~cm^{-3}$, $a_{He} = 1.0$ Å, $\tau = 0.2$, $T = 18,000$ K, $C(H\beta) = 0.1$, $a_{H} = 1.0$ Å, and $\xi = 1.0 \times 10^{-4}$. The 68% confidence level is marked by the dashed lines ($\chi^{2}_{min} = 0.0$ for synthetic data). The arrow denotes the input value.
• Figure 2: Similar to figure \ref{['Syn-y']} with plots of $\chi^{2}$ versus density, helium absorption, optical depth, and temperature.
• Figure 3: Similar to figure \ref{['Syn-y']} with plots of $\chi^{2}$ versus reddening, hydrogen absorption, and neutral hydrogen fraction.
• Figure 4: Contour plot of $\chi^{2}$ versus density and temperature for the same synthetic model used in figures \ref{['Syn-y']}-\ref{['Syn-H_3panel']}. Contours are $\Delta \chi^{2}$ = {1, 2.3, 4, 6, 9}. The degeneracy between the parameters limits the precision of their determination and, therefore, of the helium abundance.
• Figure 5: $\chi^{2}$ versus abundance for synthetic data with model parameters, $y^{+}=0.08$, $n_{e} = 100~cm^{-3}$, $a_{He} = 1.0$ Å, $T = 18,000$ K, $C(H\beta) = 0.1$, $a_{H} = 1.0$ Å, and $\xi = 1.0 \times 10^{-4}$. The only difference between the two sets is the value of the optical depth: $\tau = 0.2$ for the lighter, open squares and $\tau = 2.0$ for the darker, solid circles. For the latter, the 68% confidence level is marked by the dashed lines, and the input value is denoted by the arrow. The larger optical depth allows a secondary minimum to develop at low abundance. Such behavior highlights a deficiency in the model and mitigates the reliability of galaxies exhibiting large optical depth.