Consistency of standard cosmologies using Bayesian model comparison and tension quantification

Technical Summary: Consistency of Standard Cosmologies Using Bayesian Model Comparison and Tension Quantification

Problem Statement
The standard cosmological model, $\Lambda$CDM, has long been supported by the convergence of independent constraints from the Cosmic Microwave Background (CMB), Baryon Acoustic Oscillations (BAO), and Type Ia Supernovae (SN). However, recent literature reports persistent discrepancies ("tensions") between these probes, most notably the Hubble constant ($H_0$) tension, internal Planck consistency issues (e.g., low-$\ell$ vs. high-$\ell$ splits, CMB vs. lensing), and the "curvature tension" where Planck data prefers a closed universe while BAO data favors flatness. Furthermore, there are ongoing debates regarding whether these tensions necessitate extensions to $\Lambda$CDM, such as evolving dark energy ($w_0w_a$CDM) or non-zero neutrino masses.

A critical challenge in the field is that qualitative conclusions regarding these tensions are often highly sensitive to specific data products, likelihood implementations, and data-set combinations. Different statistical metrics (evidence ratios, posterior shifts, fit-based measures) weight prior volumes and shared subspaces differently, making it difficult to determine if a tension is a robust physical signal or an artifact of analysis choices. This paper aims to provide a unified, consistent Bayesian assessment to disentangle these factors.

Methodology
The authors employ a unified Bayesian framework to assess model comparison and data-set consistency across five cosmological models: the baseline $\Lambda$CDM and four minimal extensions:

1. $\Lambda$CDM + $\sum m_\nu$ (sum of neutrino masses).
2. $\Lambda$CDM + $\Omega_K$ (spatial curvature).
3. $w$CDM (constant dark energy equation of state).
4. $w_0w_a$CDM (evolving dark energy).

Data and Likelihoods:
The analysis utilizes a comprehensive suite of data:

• CMB: Planck Public Release 3 (PR3/2018) and Planck Public Release 4 (PR4/NPIPE). The authors compare multiple high-$\ell$ likelihood implementations (Plik, CamSpec, HiLLiPoP) and low-$\ell$ likelihoods (SimAll, LoLLiPoP), alongside CMB lensing likelihoods (L3, L4).
• BAO: Data from SDSS (DR7, DR12, DR16) and the Dark Energy Spectroscopic Instrument (DESI DR2).
• SN: Type Ia supernovae from the Dark Energy Survey (DES y5), Pantheon+, and Union3.
• Distance Ladder: Local $H_0$ measurements (Riess 2020, Freedman 2020) are included for context.

Statistical Framework:
The study utilizes three pillars of Bayesian inference:

1. Parameter Estimation: Deriving posterior distributions.
2. Model Comparison: Calculating Bayesian evidence ($Z$) and log-evidence differences ($\Delta \ln Z$) to determine model preference, incorporating Occam penalties for additional parameters.
3. Tension Quantification:
   • Evidence Ratio ($R$): Compares the joint evidence of two datasets against the product of their individual evidences. A value $\ln R \ll 0$ indicates tension.
   • Suspiciousness ($S$): A likelihood-based, prior-independent metric derived from the "Occam equation." It decomposes tension into a goodness-of-fit component and an Occam penalty component. $S$ is converted to a $p$-value and "sigma" ($\sigma$) significance using the difference in Bayesian model dimensionalities ($\tilde{d}$).

The authors use CosmoPower emulators for rapid likelihood evaluation and nested sampling (PolyChord) for evidence computation.

Key Results

• Internal CMB Consistency: Updated Planck processing (PR4) systematically improves internal consistency. The tension between low-$\ell$ and high-$\ell$ multipoles, and between primary CMB and CMB lensing, is significantly reduced when moving from PR3 (Plik) to PR4 (CamSpec, HiLLiPoP). For instance, the curvature extension's preference for a closed universe, which created tension with BAO in PR3, largely vanishes with PR4 likelihoods and updated BAO data (DESI).
• Curvature Tension: The "curvature tension" (preference for $\Omega_K < 0$ in CMB vs. flatness in BAO) is identified as largely a feature of earlier Planck likelihood implementations (PR3/Plik). With PR4 and modern BAO data (DESI DR2), the tension drops below $1\sigma$, and the preference for curvature disappears.
• Dark Energy Extensions:
  • $w$CDM: Replacing $\Lambda$ with a constant $w$ does not robustly improve fits or resolve tensions; it often merely degrades the constraining power of the CMB on $H_0$ without resolving the underlying discrepancy.
  • $w_0w_a$CDM: Apparent evidence for evolving dark energy in CMB+BAO+SN combinations is highly sensitive to the specific pairing of likelihoods. Strong preference for $w_0w_a$CDM is observed only with specific combinations (e.g., Planck PR3 + DES y5). Switching to Planck PR4 and/or Pantheon+ SN data reverses this preference, restoring a moderate preference for $\Lambda$CDM.
  • DES Dovekie Update: The paper notes that a subsequent update to the DES SN likelihood ("DES Dovekie") shifts the $\Omega_m$ posterior, bringing it into agreement with Pantheon+ and further reducing tensions, confirming that claims of a required shift to $w_0w_a$CDM based on DES y5 were premature.
• Neutrino Masses: Allowing a free sum of neutrino masses ($\sum m_\nu$) tends to absorb residual shifts without introducing new inconsistencies. The authors find no robust evidence for a standalone $\tau$-driven (optical depth) discrepancy once the full likelihood context is accounted for.
• Metric Sensitivity: The study highlights that "tension" is not a single invariant quantity. Different likelihood choices (e.g., CamSpec vs. HiLLiPoP, both based on PR4) yield measurably different tension statistics (differences of $\sim 0.5\sigma$), underscoring that "PR4" does not uniquely fix the high-$\ell$ likelihood choice.

Significance and Conclusions
The paper concludes that claims requiring an update of the standard cosmological model from $\Lambda$CDM to dynamical dark energy models ($w_0w_a$CDM) are premature. The authors argue that:

1. Many reported tensions are sensitive to analysis choices (likelihood implementations, data releases) rather than being immutable properties of the data.
2. Updates in data processing (Planck PR3 $\to$ PR4) and data compilation (SDSS $\to$ DESI) have systematically improved consistency.
3. Apparent "tension relief" in extended models often results from degrading the constraining power of a dataset (increasing uncertainty) rather than achieving a genuinely improved simultaneous fit.
4. The "curvature tension" and the strongest signals for evolving dark energy are not robust across reasonable choices of data products.

The authors advocate for maintaining multiple independent analysis pipelines to quantify residual systematics and caution against over-interpreting discrepancies that may evolve as analyses mature and systematics handling is revised. They emphasize that the current statistical evidence does not strongly necessitate a departure from the $\Lambda$CDM paradigm.