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Clarifying the Hubble constant tension with a Bayesian hierarchical model of the local distance ladder
Stephen M. Feeney, Daniel J. Mortlock, Niccolò Dalmasso
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Estimates of the Hubble constant, H₀, from the distance ladder and the cosmic microwave background (CMB) differ at the ∼3-σ level, indicating a potential issue with the standard ΛCDM cosmology. Interpreting this tension correctly requires a model comparison calculation depending on not only the traditional `n-σ' mismatch but also the tails of the likelihoods. Determining the form of the tails of the local H₀ likelihood is impossible with the standard Gaussian least-squares approximation, as it requires using non-Gaussian distributions to faithfully represent anchor likelihoods and model outliers in the Cepheid and supernova (SN) populations, and simultaneous fitting of the full distance-ladder dataset to correctly propagate uncertainties. We have developed a Bayesian hierarchical model that describes the full distance ladder, from nearby geometric anchors through Cepheids to Hubble-Flow SNe. This model does not rely on any distributions being Gaussian, allowing outliers to be modeled and obviating the need for arbitrary data cuts. Sampling from the ∼3000-parameter joint posterior using Hamiltonian Monte Carlo, 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. Our high-fidelity sampling of the low-H₀ tail of the distance-ladder likelihood allows us to apply Bayesian model comparison to assess the evidence for deviation from ΛCDM. We set up this comparison to yield a lower limit on the odds of the underlying model being ΛCDM given the distance-ladder and Planck XIII (2016) CMB data. The odds against ΛCDM are at worst 10:1 or 7:1, depending on whether the SNe outliers are cut or modeled, or 60:1 if an approximation to the Planck Int. XLVI (2016) likelihood is used.
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