Clarifying the Hubble constant tension with a Bayesian hierarchical model of the local distance ladder

Abstract

Estimates of the Hubble constant, H0, from the local distance ladder and from the cosmic microwave background (CMB) are discrepant at the ∼3-σ level, indicating a potential issue with the standard ΛCDM cosmology. A probabilistic (i.e., Bayesian) interpretation of this tension requires a model comparison calculation, which in turn depends strongly on the tails of the H₀ likelihoods. Evaluating the tails of the local H₀ likelihood requires the use of non-Gaussian distributions to faithfully represent anchor likelihoods and outliers, and simultaneous fitting of the complete distance ladder dataset to ensure correct uncertainty propagation. We have hence developed a Bayesian hierarchical model of the full distance ladder that does not rely on Gaussian distributions and allows outliers to be modelled without arbitrary data cuts. Marginalizing over the full ∼3000-parameter joint posterior distribution we find H0 = (72.72 ± 1.67) km s¯¹ Mpc¯¹ when applied to the outlier-cleaned Riess et al. (2016) data, and (73.15 ± 1.78) km s¯¹ Mpc¯¹ with SN outliers reintroduced (the pre-cut Cepheid dataset is not available). Using our precise evaluation of the tails of the H0 likelihood, we apply Bayesian model comparison to assess the evidence for deviation from ΛCDM given the distance-ladder and CMB data. The odds against ΛCDM are at worst ∼10:1 when considering the Planck Collaboration (2016b) data, regardless of outlier treatment, considerably less dramatic than naïvely implied by the 2.8-σ discrepancy. These odds become ∼60:1 when an approximation to the more-discrepant Planck Collaboration (2016c) likelihood is included.