Clarifying the Hubble constant tension with a Bayesian hierarchical model of the local distance ladder (1707.00007v2)

Abstract: Estimates of the Hubble constant, $H_0$, from the distance ladder and the cosmic microwave background (CMB) differ at the $\sim$3-$\sigma$ level, indicating a potential issue with the standard $\Lambda$CDM cosmology. Interpreting this tension correctly requires a model comparison calculation depending on not only the traditional `$n$-$\sigma$' mismatch but also the tails of the likelihoods. Determining the form of the tails of the local $H_0$ 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 $\sim$3000-parameter joint posterior using Hamiltonian Monte Carlo, we find $H_0$ = (72.72 $\pm$ 1.67) ${\rm km\,s^{-1}\,Mpc^{-1}}$ when applied to the outlier-cleaned Riess et al. (2016) data, and ($73.15 \pm 1.78$) ${\rm km\,s^{-1}\,Mpc^{-1}}$ with SN outliers reintroduced. Our high-fidelity sampling of the low-$H_0$ tail of the distance-ladder likelihood allows us to apply Bayesian model comparison to assess the evidence for deviation from $\Lambda$CDM. We set up this comparison to yield a lower limit on the odds of the underlying model being $\Lambda$CDM given the distance-ladder and Planck XIII (2016) CMB data. The odds against $\Lambda$CDM 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.

• The paper introduces a Bayesian hierarchical model to fit the entire distance ladder data while robustly handling outliers and non-Gaussian error distributions.
• The paper reports an H0 posterior of 72.72 ± 1.67 km/s/Mpc, with slightly higher values when incorporating SNe outliers using heavier-tailed distributions.
• The paper compares ΛCDM with an alternative model, reducing biases from Gaussian assumptions and tempering the interpretation of new physics.

Bayesian Hierarchical Modeling of the Hubble Constant Discrepancy

This paper by Feeney et al. addresses the long-standing tension between local determinations of the Hubble constant ($H_0$) and values inferred from cosmic microwave background (CMB) measurements under the standard $\Lambda$CDM cosmological model. Local measurements, primarily relying on the distance ladder comprising Cepheids and supernovae (SNe), suggest a higher $H_0$ than the extrapolations from the CMB data. This discrepancy has been a subject of extensive discussion as it may indicate new physics beyond the standard cosmological model or unaccounted-for systematic errors in the measurements.

The authors employ a Bayesian hierarchical model (BHM) to quantify the Hubble constant from local data while directly addressing the statistical issues related to outlier treatment and likelihood non-Gaussianity, which are often overlooked in conventional analyses. By employing this framework, the authors ensure a self-consistent fit to the entire distance ladder data, mitigating some of the potential biases inherent in traditional least-squares regression methods, which typically estimate parameters separately for Cepheids and SNe.

One significant strength of this BHM approach is its ability to incorporate all relevant uncertainties systematically and handle non-Gaussian errors, especially in the presence of outliers, by using heavier-tailed distributions like the Student-t distribution. This is particularly pertinent as accurate assessment of the full posterior distribution of $H_0$ requires realistic modeling of tails, rather than assuming Gaussianity.

In their analysis, employing Hamiltonian Monte Carlo sampling, the authors find that the BHM-derived $H_0$ aligns closely with conventional methods but with additional robustness in characterizing the tail behavior. They report an $H_0$ posterior of (72.72 ± 1.67) km/s/Mpc when using the outlier-cleaned dataset, which shifts slightly higher when SNe outliers are incorporated using a model that accounts for them rather than excluding them, yielding (73.15 ± 1.78) km/s/Mpc.

Interestingly, the authors extend the framework to include model comparison between $\Lambda$CDM and an alternative model allowing discrepancies in $H_0$ and the deceleration parameter $q_0$ inferred locally and from Planck CMB data. This analysis highlights the reduced interpretive bias present in naively assuming Gaussian residuals or directly comparing high-significance sigma deviations. The result is a less dramatic assessment of the likelihood of new physics than traditional $p$-values suggest, aligning with a Bayesian evidence-based approach.

The implications of this research lie primarily in the methodological advancements it affords to cosmological parameter estimation. By employing a BHM, the authors provide a richer statistical treatment of the Hubble constant data, thus enhancing confidence in interpreting the outputs of the local distance ladder and their implications for cosmology. The approach also underscores the complexity of $H_0$ tension and invites further integration of hierarchical modeling with other cosmological datasets to comprehensively address and potentially resolve this critical discrepancy in modern cosmology. Future work could explore integrating Gaia parallax data to further refine distance ladder estimates or extending the model to a broader redshift range to better probe potential physics beyond the standard model.