Shades of dark uncertainty and consensus value for the Newtonian constant of gravitation

Abstract

The Newtonian constant of gravitation, G, stands out in the landscape of the most common fundamental constants owing to its surprisingly large relative uncertainty, which is attributable mostly to the dispersion of the values measured for it by different methods and in different experiments, each of which may have rather small relative uncertainty. This study focuses on a set of measurements of G comprising results published very recently as well as older results, some of which have been corrected since the original publication. This set is inconsistent, in the sense that the dispersion of the measured values is significantly larger than what their reported uncertainties suggest that it should be. Furthermore, there is a loosely defined group of measured values that lie fairly close to a consensus value that may reasonably be derived from all the measurement results, and then there are one or more groups with measured values farther away from the consensus value, some appreciably higher, others lower. This same general pattern is often observed in many other interlaboratory studies and meta-analyses. In the conventional treatments of such data, the mutual inconsistency is addressed by inflating the reported uncertainties, either multiplicatively, or by the addition of 'random effects', both reflecting the presence of dark uncertainty. The former approach is often used by CODATA and by the Particle Data Group, and the latter is common in medical meta-analysis and in metrology. However, both achieve consistency ignoring how the measured values are arranged relative to the consensus value, and measured values close to the consensus value often tend to be penalized excessively, by such 'extra' uncertainty. We propose a new procedure for consensus building that models the results using latent clusters with different shades of dark uncertainty, which assigns a customized amount of dark uncertainty to each measured value, as a mixture of those shades, and does so taking into account both the placement of the measured values relative to the consensus value, and the reported uncertainties. We demonstrate this procedure by deriving a new estimate for G, as a consensus value G = 6.67408 × 10-11 m3 kg-1 s-2, with μ(G) = 0.00024 × 10-11 m3 kg-1 s-2