A Robust Bayesian Meta-Analysis for Estimating the Hubble Constant via Time Delay Cosmography

We propose a Bayesian meta-analysis to infer the current expansion rate of the Universe, called the Hubble constant ($H_0$), via time delay cosmography. Inputs of the meta-analysis are estimates of two properties for each pair of gravitationally lensed images; time delay and Fermat potential difference estimates with their standard errors. A meta-analysis can be appealing in practice because obtaining each estimate from even a single lens system involves substantial human efforts, and thus estimates are often separately obtained and published. Moreover, numerous estimates are expected to be available once the Rubin Observatory starts monitoring thousands of strong gravitational lens systems. This work focuses on combining these estimates from independent studies to infer $H_0$ in a robust manner. The robustness is crucial because currently up to eight lens systems are used to infer $H_0$, and thus any biased input can severely affect the resulting $H_0$ estimate. For this purpose, we adopt Student's $t$ error for the input estimates. We investigate properties of the resulting $H_0$ estimate via two simulation studies with realistic imaging data. It turns out that the meta-analysis can infer $H_0$ with sub-percent bias and about 1% level of coefficient of variation, even when 30% of inputs are manipulated to be outliers. We also apply the meta-analysis to three gravitationally lensed systems to obtain an $H_0$ estimate and compare it with existing estimates. An R package, h0, is publicly available for fitting the proposed meta-analysis.