Consistency of standard cosmologies using Bayesian model comparison and tension quantification

TL;DR

The paper presents a unified Bayesian framework to assess model consistency and data-set tension for LCDM and minimal extensions using CMB, BAO, and SN data, leveraging Bayesian evidence $ abla olinebreak olimits Z$, Bayes factors, and tension diagnostics $ abla olinebreak olimits R$ and $ abla olinebreak olimits S$ across Planck PR3/PR4, DESI, Pantheon+, and DESy5 inputs. It demonstrates that updated Planck processing improves internal CMB consistency, curvature tension is largely a PR3-era feature, and evolving dark energy claims depend on SN likelihoods, with DES Dovekie bringing SN constraints in line with Pantheon+. Neutrino-mass extensions neither strongly improve the fit nor overturn the Occam penalty, while the data do not provide robust evidence for a required shift from LCDM to $w_0w_a$CDM. The study highlights the importance of consistent Bayesian diagnostics and multiple data-products, arguing for cautious interpretation of tensions and beyond-LCDM claims until end-to-end, cross-validated analyses with diverse likelihoods and data sets are available.

Abstract

We present a unified Bayesian assessment of model comparison and data-set consistency for LCDM (cold dark matter plus a cosmological constant) and minimal extensions (neutrino mass, spatial curvature, constant or evolving dark energy) using cosmic microwave background (CMB), baryon acoustic oscillation (BAO), and type Ia supernova (SN) data. The major results are summarized in the first three figures. We quantify model preference with Bayesian evidence and assess consistency with complementary evidence- and likelihood-based diagnostics applied uniformly across data-set combinations. For the models considered, updated Planck processing systematically improves internal CMB consistency (low-$\ell$ versus high-$\ell$, and primary CMB versus CMB lensing). The preference for a closed geometry and an associated ``curvature tension'' with BAO and/or CMB lensing are largely confined to earlier Planck likelihood implementations and weaken substantially when using updated CMB processing and more recent BAO measurements. Apparent evidence for evolving dark energy in CMB+BAO+SN combinations depends sensitively on the specific pairing of CMB and SN likelihoods: plausible alternatives shift inferred tensions by more than $1\,σ$ and can completely reverse the preferred model. Allowing a free neutrino mass tends to absorb residual shifts without introducing new inconsistencies, and we do not find robust evidence for a standalone $τ$-driven discrepancy once the full likelihood context is accounted for. We conclude that claims of a required update of our standard cosmological model from LCDM to $w_0w_a$CDM are premature.

Figures (16)

• Figure 1: Data-set tension based on the Bayesian evidence ratio statistic $\mathcal{R}$ from \ref{['eq:R']} for a range of data-set combinations as labeled on the left-hand side. The black points correspond to the $\Lambda\mathrm{CDM}$ model and the colored points to its extensions with neutrinos (blue), curvature (green), constant (orange), and dynamical (red) dark energy. Symbols with different shapes highlight different CMB likelihoods. The error bars reflect the sampling uncertainty. A value $\ln\mathcal{R}\ll0$ indicates tension, with the orange shaded regions referring to the Jeffrey's scale of moderate, strong, and very strong tension. $\ln\mathcal{R}\gg0$ indicates agreement.
• Figure 2: Data-set tensions based on the average likelihood-ratio statistic $\mathcal{S}$ from \ref{['eq:S']} for a range of data-set combinations as labeled on the left-hand side, expressed as a $p$-value (top axis) and numbers of $\sigma$ (bottom axis). Repeated "vs" implies 3-way or 4-way comparisons according to \ref{['eq:multi_tension']}. The black points correspond to the $\Lambda\mathrm{CDM}$ model and the colored points to its extensions with neutrinos (blue), curvature (green), constant dark energy (orange), and dynamical dark energy (red). Symbols with different shapes highlight different CMB likelihoods. The error bars reflect the sampling uncertainty. Some error bars appear surprisingly big, which is an effect of the translation from $\mathcal{S}$ to a $p$-value using \ref{['eq:tension_sigma']} for a vanishing dimensionality $\tilde{\mathcal{d}}$, i.e., when $\mathcal{d}_A + \mathcal{d}_B \approx \mathcal{d}_{AB}$.
• Figure 3: Model comparisons relative to $\Lambda\mathrm{CDM}$ in terms of Bayesian evidence ratios (see \ref{['eq:model_comparison']}) based on different data-set combinations. The black points correspond to the $\Lambda\mathrm{CDM}$ model and the colored points to its extensions with neutrinos (blue), curvature (green), constant dark energy (orange), and dynamical dark energy (red). The error bars reflect the sampling uncertainty. The shaded regions illustrate the Jeffreys' scale, categorizing the extensions as weakly, strongly, or very strongly favored compared to $\Lambda\mathrm{CDM}$ in green, or disfavored in orange.
• Figure 4: Example of how increasing the parameter space from $\Lambda\mathrm{CDM}$ (where $w=-1$) to $w\mathrm{CDM}$ relieves the Hubble tension. However, this is not because both observables agree on $H_0$, but because the CMB loses almost all constraining power on $H_0$ when the parameter space is opened up.
• Figure 5: $\Lambda\mathrm{CDM}+\Omega_K$ posteriors for various data sets. Note the strong preference for closed universes under PR3, which gets significantly reduced when switching to PR4, making CMB data more compatible with the results from BAOs.