Evidence for extra radiation? Profile likelihood versus Bayesian posterior

TL;DR

The paper compares Bayesian marginalised posteriors and profile likelihoods for the effective number of relativistic species $N_{ m eff}$ to test claims of extra radiation. It shows a measurable, but modest, tension between the two constraint methods stemming from a volume effect linked to the CMB’s indirect sensitivity to $N_{ m eff}$ via $z_{ m eq}$. Including external information on the Hubble parameter reduces this discrepancy, indicating the difference is not the sole driver of the inferred excess radiation. The study concludes that the preference for $ ext{excess radiation}$ is real and not an artifact of the statistical construction, while acknowledging potential residual systematics that could influence the result.

Abstract

A number of recent analyses of cosmological data have reported hints for the presence of extra radiation beyond the standard model expectation. In order to test the robustness of these claims under different methods of constructing parameter constraints, we perform a Bayesian posterior-based and a likelihood profile-based analysis of current data. We confirm the presence of a slight discrepancy between posterior- and profile-based constraints, with the marginalised posterior preferring higher values of the effective number of neutrino species N_eff. This can be traced back to a volume effect occurring during the marginalisation process, and we demonstrate that the effect is related to the fact that cosmic microwave background (CMB) data constrain N_eff only indirectly via the redshift of matter-radiation equality. Once present CMB data are combined with external information about, e.g., the Hubble parameter, the difference between the methods becomes small compared to the uncertainty of N_eff. We conclude that the preference of precision cosmological data for excess radiation is "real" and not an artifact of a specific choice of credible/confidence interval construction.

Figures (4)

• Figure 1: Constraints on $N_{\rm eff}$ from WMAP7+ACT data. Thin black lines denote the posterior probability density marginalised over the other parameter directions for three different choices of prior (solid: $H_0$, dashed: $\theta_{\rm s}$, dotted: $\Omega_{\Lambda}$). The profile likelihood is plotted in thick red lines, both in terms of $\mathcal{L}^{\rm p}/\mathcal{L}^{\rm p}_{\rm max}$ (solid) and $\Delta \chi^2_{\rm eff}$ (dotted).
• Figure 2: Variance of the marginalised posterior probability of $\omega_{m}$ on slices of width $\Delta N_{\rm eff} = 1$ as a function of $N_{\rm eff}$. The crosses mark the values extracted from the Markov chains, the red line is the prediction based on the variance of $z_{\rm eq}$, consisting of a constant term induced by the bin width (equation (\ref{['eq:binvar']}), dashed line) and the intrinsic variance of $\omega_{\rm m}$ (equation (\ref{['eq:intvar']}), dotted line).
• Figure 3: Same as figure \ref{['fig:wmapact']}, for WMAP7+ACT+HST.
• Figure 4: Probability of finding at least one sample within $\Delta \chi^2_{\rm eff} = x$ of the true maximum of the $n$-dimensional Gaussian posterior $\mathcal{P}$, if the Markov chain was generated at a temperature $T$ and contains $N$ independent samples. Top left: dependence on $N$ and $n$ for $T=1$ and $x = 0.2$. Top right: dependence on $N$ and $T$ for $n=10$ and $x = 0.2$. Bottom: dependence on $N$ and $x$ for $T=1$ and $n = 10$.