Systematic assessment of the Hubble tension via Bayesian jackknife testing

TL;DR

We address the Hubble tension by applying a Bayesian jackknife approach (CHIBORG) to 16 independent measurements of $H_0$, comparing early- vs late-Universe determinations and exploring phenomenological bias scenarios. We develop a hierarchical bias model and use PolyChord to compute evidences for competing hypotheses, yielding a model-weighted posterior for $H_0$. The results show a preference for late-Universe biases but no single scenario dominates, with a multi-modal $H_0$ posterior spanning $66.7 < H_0 < 72.7$ km/s/Mpc at 95\% credibility. The work highlights the need for improved correlation modeling and physically motivated bias mappings, and provides a public codebase to enable further exploration.

Abstract

Statistically-significant differences in the value of the Hubble parameter are found depending on the measurement method that is used, a result known as the Hubble tension. A variety of ways of comparing, grouping, and excluding measurements have been used to try to explain this, either in terms of physical effects or systematic errors. We present a systematic 'Bayesian jackknife' analysis of 16 independent measurements of the Hubble parameter in an attempt to identify whether the measurements fall into meaningful clusters that would help explain the origin of the tension. After evaluating evidence ratios for the commonly-used split into early- vs late-time measurements, we then study a range of simplified alternative physical scenarios that reflect different physical origins of an apparent bias or shift in the value of $H_0$, assigning phenomenological population parameters to each subset. These include scenarios where specific subsets are biased (e.g. due to unrecognised experimental systematics in the local distance ladder or cosmic microwave background measurements), as well as more cosmologically-motivated cases involving modifications to the expansion history. Many of these scenarios have similar marginal likelihood, but the model where no measurements are biased is strongly disfavoured. Finally, we marginalise over all these scenarios to estimate the 'model agnostic' posterior distribution of $H_0$. The resulting distribution is mildly multi-modal, but modestly favours values near $H_0=68$ km/s/Mpc, with a 95\% credible region of $66.7 < H_0 < 72.7$ km/s/Mpc.

Figures (8)

• Figure 1: Parameter structure used in the initial model comparison, illustrating the relationships between hyperparameters, model parameters, and the data vector $\mathbf{d}$. Parameters shown in green are fixed, while those in red are marginalised over in the analysis. Parameters shown in dashed green lines are sometimes fixed, depending on the test. In later phenomenological scenarios, $\bar{\theta}_\mu$ and $\bar{\theta}_C$ are removed, with $\mu_\varepsilon$ and $C_\varepsilon$ fixed throughout the analysis.
• Figure 2: The 16 independent $H_0$ measurements used in this analysis, corresponding to the values listed in Table \ref{['tab:16 Hubble measurements']}. Early Universe measurements (CMB and BAO) are shown in blue, late Universe distance ladder measurements in red, and one-step methods with open red markers. The shaded bands indicate the $\pm 1\sigma$ regions for the Planck CMB (blue) and Cepheids-SN Ia (red) measurements, which are commonly used as reference values in the Hubble tension discussion.
• Figure 3: Differences in the cosmic expansion rate ($\Delta H(z)$) and angular diameter distance $D_A(z)$ relative to a reference $\Lambda$CDM cosmology (black dashed line), for dark energy models with the same value of $H(z=0)$. Coloured curves correspond to models with varying dark energy parameters ($w_0$, $w_a$), while keeping all other parameters fixed. Grey dotted lines show the effect of taking the $\Lambda$CDM model and shifting $H_0$ in increments of 2 sM, providing a direct mapping between deviations in $H(z)$ at intermediate redshifts and their equivalent bias in the inferred $H_0$.
• Figure 4: Bias specifications used in the analysis (as shown in Table \ref{['tab: bias specs']}, grouped by measurement class: LDL, Local One-step, Distant One-step, BAO, and CMB. Each point represents the bias mean $\mu_\varepsilon$, with error bars corresponding to the bias uncertainty. For parameters expressed in terms of the reported measurement uncertainties ($\sigma$) in table, the plotted value corresponds to the median of the reported uncertainty range. While the plot shows only these representative medians for clarity, the full analysis used the complete set of reported measurements and their uncertainties.
• Figure 5: Posterior support for early versus late bias scenarios under two definitions of the early/late division. The $z \geq 2$ split (shown in blue) isolates the CMB as early, with all other probes treated as late, while the $z \geq 0.2$ split (shown in red) assigns local probes (LDL and low-$z$ one-step) to the early category and higher-redshift probes to the late category. In both cases, late bias is favoured, though the strength of this preference depends on the placement of the cut-off.