$\texttt{unimpeded}$: A Public Grid of Nested Sampling Chains for Cosmological Model Comparison and Tension Analysis

TL;DR

The paper introduces unimpeded, a public Python library and Zenodo data repository of pre-computed nested sampling and MCMC chains to enable robust Bayesian model comparison and tension analysis in cosmology. It implements a grid of $8$ cosmological models and $39$ datasets (including their pairwise combinations) to systematically assess model preference and cross-dataset tensions, using Bayesian evidence and five complementary tension metrics. Key findings show that the base $\Lambda$CDM model is most frequently preferred when combining datasets, with significant tensions such as DES–Planck ($\sim3.57\sigma$) and SH0ES–Planck ($\sim3.27\sigma$) within $\Lambda$CDM; the $S_8$ tension is high-dimensional and mitigated in extended models, while the Hubble tension persists in low-dimensional form. The data products enable reproducible cosmological analyses, illustrate the robustness of $\Lambda$CDM against current data, and pave the way for rapid, model-aware analyses with future datasets and method extensions.

Abstract

Bayesian inference is central to modern cosmology, yet comprehensive model comparison and tension quantification remain computationally prohibitive for many researchers. To address this, we release $\texttt{unimpeded}$, a publicly available Python library and data repository providing pre-computed nested sampling and MCMC chains. We apply this resource to conduct a systematic analysis across a grid of eight cosmological models, including $Λ$CDM and seven extensions, and 39 datasets, including individual probes and their pairwise combinations. Our model comparison reveals that whilst individual datasets show varied preferences for model extensions, the base $Λ$CDM model is most frequently preferred in combined analyses, with the general trend suggesting that evidence for new physics is diluted when probes are combined. Using five complementary statistics, we quantify tensions, finding the most significant to be between DES and Planck (3.57$σ$) and SH0ES and Planck (3.27$σ$) within $Λ$CDM. We characterise the $S_8$ tension as high-dimensional ($d_G=6.62$) and resolvable in extended models, whereas the Hubble tension is low-dimensional and persists across the model space. Caution should be exercised when combining datasets in tension. The $\texttt{unimpeded}$ data products, hosted on Zenodo, provide a powerful resource for reproducible cosmological analysis and underscore the robustness of the $Λ$CDM model against the current compendium of data.

Figures (13)

• Figure 1: Evolution of the live point count throughout nested sampling runs for $\Lambda$CDM and $w_0w_a$CDM models with Planck with CMB lensing and Pantheon datasets. Three distinct phases are visible: (1) initial compression where the first $n_{\text{prior}} - n_{\text{live}} \approx 9{,}000$ iterations sequentially delete the lowest-likelihood points without replacement, reducing the live point count from $n_{\text{prior}} \approx 10^4$ to the target value $n_{\text{live}} \approx 10^3$, which will enter the main nested sampling stage; (2) main sampling phase where the live point count oscillates above $n_{\text{live}}$ due to synchronous parallel processing with 760 cores; (3) final deletion phase where the live point count decreases from $n_{\text{live}}$ to zero as the remaining points are systematically removed one by one. The iteration number $i$ ($x$-axis) maps to the compressed log-prior volume via $\langle \log X_i \rangle = -\sum_{k=1}^{i} 1/n_k$, where $n_k$ is the live point count at iteration $k$, as shown by the $y$-axis (see \ref{['ssec:dynamic_ns']} for details) Hu2023aeons. The different termination points reflect the different Kullback--Leibler divergences between prior and posterior for each model-dataset combination. More complex models like $w_0w_a$CDM (green) require slightly more iterations than simpler models like $\Lambda$CDM (orange), but this effect is not as dominant as the variation across different datasets, with Planck with CMB lensing (blue) requiring substantially more iterations than Pantheon (orange, green).
• Figure 2: Corner plot showing posterior distributions for the $\Omega_k\Lambda$CDM cosmological model constrained by Planck with CMB lensing + SDSS data. The diagonal panels show the one-dimensional marginalised prior (blue) and Planck with CMB lensing + SDSS posterior (orange) distributions, demonstrating the constraining power of the observational data. The lower triangular panels display the two-dimensional joint posterior and prior, where the inner (darker blue and darker orange) and outer (lighter blue and lighter orange) contours correspond to the 68% ($1\sigma$) and 95% ($2\sigma$) credible regions, respectively. The upper triangular panels show scatter plots of samples drawn from the posterior, visually representing parameter correlations. The posterior volume (orange) is much smaller than the prior volume (blue). This corner plot was created using anesthetic.
• Figure 3: Heatmap of the log-posterior model probabilities, $\log \text{P}(\mathcal{M}_i|D)$, for each cosmological model ($x$-axis) evaluated against individual dataset ($y$-axis). Bluer colours indicate stronger statistical support for a model given the data. Comparison should only be made horizontally across models for a fixed dataset, as the sum of evidences in \ref{['eq:model_prob']} is taken over all models for that specific dataset. The final column (in yellow) shows the normalising factor $\log\left(\sum_j \mathcal{Z}_j\right)$, the logarithm of the denominator of \ref{['eq:model_prob']}. The raw log-evidence for any model-dataset combination can be recovered by multiplying the probability value in that cell by the normalising factor of that row. The results show that while different datasets favour different model extensions, the base $\Lambda$CDM model emerges as the most consistently well-performing model across all individual datasets (overall blue).
• Figure 4: Same as \ref{['fig:model_comp_single']}, but for combinations of datasets. The final column (in yellow) shows the normalising factor $\log\left(\sum_j \mathcal{Z}_j\right)$, the logarithm of the denominator of \ref{['eq:model_prob']}, allowing recovery of raw log-evidence values by multiplying the probability in each cell by the normalising factor of that row. The combination of multiple probes sharpens the model comparison, further strengthening the preference for $\Lambda$CDM and increasing the degree to which extended models are disfavoured. Part 1 of combined datasets.
• Figure 5: Same as \ref{['fig:model_comp_single']}, but for combinations of datasets. The final column (in yellow) shows the normalising factor $\log\left(\sum_j \mathcal{Z}_j\right)$, the logarithm of the denominator of \ref{['eq:model_prob']}, allowing recovery of raw log-evidence values by multiplying the probability in each cell by the normalising factor of that row. The combination of multiple probes sharpens the model comparison, further strengthening the preference for $\Lambda$CDM and increasing the degree to which extended models are disfavoured. Part 2 of combined datasets.