Cosmological Tensions and Robustness Tests

A cosmological tension is a reproducible failure of two or more specified datasets to fit or predict one another within a declared model—not simply a difference between two headline means. Robustness requires the same parameter definitions, likelihood versions, covariance, calibration, selection, priors, and multiplicity accounting, followed by tests that were not chosen to maximize the discrepancy. The public verdict through 2026-08-19 is mixed: several important discrepancies and hints remain, but their strength and interpretation depend materially on probe, release, model, and systematic assumptions.

Required background. Probability and Statistical Inference supplies joint distributions, calibrated test statistics, posterior prediction, and multiple-testing corrections.

Helpful background. Einstein Equations supplies the gravitational baseline; Relativistic Matter and Compact Objects supplies source and foreground systematics; Relativistic Cosmology supplies the probes and model degeneracies; and Computational Relativity supplies simulation and convergence checks.

Define the discrepancy before quoting significance

Suppose two analyses estimate the same $k$-component parameter vector. Let $\boldsymbol\mu_A$ and $\boldsymbol\mu_B$ denote the estimators, and let $\mathsf C_A$, $\mathsf C_B$, and $\mathsf C_{AB}$ denote their repeated-sampling covariance and cross-covariance under the null. In a Gaussian approximation,

$$\boldsymbol\Delta=\boldsymbol\mu_A-\boldsymbol\mu_B,$$ $$\mathsf C_\Delta =\mathsf C_A+\mathsf C_B -\mathsf C_{AB}-\mathsf C_{AB}^{\mathsf T}, \qquad T=\boldsymbol\Delta^{\mathsf T} \mathsf C_\Delta^{-1}\boldsymbol\Delta.$$

If the difference has zero mean under the null, is approximately Gaussian, and has the stated nonsingular sampling covariance, $T$ has a $\chi_k^2$ reference distribution for $k$ prespecified identifiable directions. Estimating the covariance, fitting additional choices, or restricting to a singular subspace requires a corresponding change in the calibration. Posterior covariance matrices can stand in for these sampling covariances only in a regular Gaussian, weak-prior regime where the prior does not appreciably couple or shrink the two estimates. With an informative prior reused in both analyses, inserting two posterior covariances effectively counts that prior twice and $T$ is not generically $\chi_k^2$; use a joint difference posterior or simulation calibration instead. The cross-covariance is essential: common sky modes, calibrators, simulations, or external priors can correlate the estimates. A one-dimensional “difference in means divided by quadrature errors” is valid only when the quantities, models, covariance interpretation, and independence assumptions match.

Shared errors have a different effect on a discrepancy than on a combined mean. For two equal-variance measurements with correlation $\rho$, $\operatorname{Var}(\mu_A-\mu_B)=2\sigma^2(1-\rho)$, whereas the variance of their average is $\sigma^2(1+\rho)/2$. A common calibration offset cancels from a difference but cannot be averaged away. Positive correlation therefore makes a fixed difference more significant while making the combined mean less precise; using quadrature errors in both problems misses this distinction.

Non-Gaussian posteriors, hard prior boundaries, weakly identified extensions, or data-dependent choices invalidate the naive conversion to a Gaussian number of standard deviations. In those cases, draw parameter differences from a joint posterior or calibrate a prespecified discrepancy on end-to-end simulations. Report both the statistic and the reference distribution used to turn it into a probability.

Local, global, and predictive questions differ

A local probability assumes that the parameter, dataset split, redshift range, scale range, and model extension were specified before looking. If the largest discrepancy was selected from many correlated possibilities, the global probability must include that search. For $m$ independent tests with local tail probability $p_{\rm loc}$, a simple illustration is

$$p_{\rm glob}=1-(1-p_{\rm loc})^m,$$

though real cosmological tests are correlated and normally require simulation.

A parameter-difference test asks whether two posterior summaries overlap as expected. A posterior predictive test instead asks whether $D_A$ predicts $D_B$. If $\boldsymbol\lambda$ contains every shared nuisance and the datasets are conditionally independent given $(\boldsymbol\theta,\boldsymbol\lambda)$,

$$p(D_B\mid D_A,M) =\int d\boldsymbol\theta\,d\boldsymbol\lambda\, p(D_B\mid\boldsymbol\theta,\boldsymbol\lambda,M) p(\boldsymbol\theta,\boldsymbol\lambda\mid D_A,M).$$

Residual cross-dependence requires the corresponding joint conditional rather than this factorization.

The two can disagree because a broad degeneracy may hide a poor fit in data space, or a prior may make parameter posteriors appear separated despite adequate predictions. Both should be reported when feasible.

A robustness ladder

1. Exact replication. Reproduce each separate likelihood and posterior with the named release, model, base parameters, priors, nuisance treatment, selections, code, and convergence criteria.
2. Internal splits and nulls. Repeat by sky region, observing epoch, instrument frequency, object population, redshift, and scale, using covariance between splits and a multiplicity plan.
3. Calibration and selection stress tests. Vary zero points, distance anchors, beam or transfer functions, photometric-redshift calibration, completeness, foregrounds, and cuts within independently justified uncertainties.
4. Likelihood and prior alternatives. Replace approximations, broaden nuisance models, transform parameter bases with Jacobians, and inspect boundary dependence.
5. Model extensions. Require an extension to predict the discrepant observables jointly. Merely widening a posterior or adding a parameter that absorbs residuals is not a resolution.
6. Out-of-sample confirmation. Test a fixed prediction on an independent calibration, new release, or withheld split. This is the strongest step because it limits after-the-fact flexibility.

The order prevents a common mistake: jumping from a marginal parameter discrepancy to new physics before verifying the data and statistical model. It also prevents the opposite mistake of invoking unspecified systematics without showing that a plausible nuisance shift can explain the residual pattern.

Results and limitations through 2026-08-19

The evidence status at 2026-08-19 does not support one universal “crisis” verdict. The competing interpretations remain new physics, underestimated systematics, calibration or selection shifts, likelihood or prior effects, unmodeled covariance, and statistical fluctuation.

Evidence or result	Current implication	Scope, limitation, or unresolved issue
Expansion-rate comparisons	Planck-conditioned and local-distance-ladder inferences remain notably separated, while several other methods lie nearer one side or between them	Cortês, Lahav, and Liddle 2026, §25.3.1 stresses both possible new physics and anchor/calibration systematics; values are model and ladder dependent
Weak-lensing amplitude	Earlier low-$S_8$ indications have weakened substantially in more recent survey analyses	Cortês, Lahav, and Liddle 2026, §25.3.6 reports that the former discrepancy has essentially disappeared; shear, redshift, intrinsic-alignment, baryon, and nonlinear choices remain relevant
DESI DR2 plus CMB and supernovae	Time-varying dark-energy fits can improve over $\Lambda$CDM, but the size changes with the supernova sample and is smaller without supernovae	Abdul Karim et al. 2025, §VII and Tables V–VI is conditional on the $w_0w_a$ form, priors, CMB likelihood, BAO compression, and supernova calibration
ACT DR6 CMB spectra	An independent high-resolution CMB analysis finds internally consistent spectra and broad agreement with Planck-conditioned $\Lambda$CDM parameters	Louis et al. 2025, §§7–8 and Appendix H binds the statement to DR6.02 masks, transfer cuts, calibration, foregrounds, covariance, and data combination
Planck 2018 internal and external checks	Base-$\Lambda$CDM fits many CMB observables, while selected anomalies and late-time comparisons motivate further tests	Planck Collaboration 2020, §§2–6 is a fully specified 2018-data case; extension posteriors and significances inherit its likelihood and priors
End-to-end null simulations	Calibrate split tests, maximum-discrepancy searches, coverage, and nuisance recovery under controlled truth	A null result has low power if the simulated systematic class is too narrow; an unmodeled real effect can evade every injected test

The cross-probe synthesis in Cortês, Lahav, and Liddle 2026, §§25.4–25.5 judges $\Lambda$CDM an excellent broad description while treating the Hubble and evolving-dark-energy indications as not securely established. That is a dated assessment, not evidence that future discrepancies must disappear.

Test the competing explanations

New physics should improve a joint likelihood and make additional predictions, not merely move one derived parameter. Underestimated systematics should correspond to a physically plausible calibration, selection, foreground, or modeling perturbation whose required size is compatible with independent controls. Likelihood or prior effects are diagnosed by changing approximations, parameter bases, and prior measures without changing the data. Dataset covariance is tested through overlap maps, common simulations, and joint rather than product likelihoods. Statistical fluctuation is assessed with a global, simulation-calibrated distribution fixed before seeing the anomaly.

Expanded nuisance models are informative only if constrained by calibration data or meaningful priors. An arbitrarily flexible nuisance can erase any discrepancy. Likewise, an arbitrary cosmological extension can make contours overlap by increasing uncertainty. A convincing explanation recovers the detailed residual pattern and survives leave-one-split-out prediction.

Three complementary cross-checks

First compute a parameter-difference statistic with the full cross-covariance and repeat it in more than one physically meaningful basis. A change caused solely by a nonlinear summary warns that the Gaussian approximation is inadequate.

Second perform posterior predictive checks in data space. Draw parameters from probe $A$, simulate probe $B$ with its response, selection, and nuisances, and compare a prespecified residual vector or goodness-of-fit statistic. Reverse the direction because $A$ predicting $B$ is not equivalent to $B$ predicting $A$.

Third perform leave-one-split-out tests: omit one instrument band, sky region, redshift interval, object class, or calibration anchor, refit without inspecting its answer, and predict the omitted block. Record all attempted splits. This localizes a disagreement without rewarding the most discrepant split after the fact.

Exercise: a local discrepancy after a search

An analysis inspects four independent, prespecified-in-form dataset splits and reports the largest two-sided local discrepancy as $2\sigma$, for which $p_{\rm loc}\simeq0.0455$. Under the simplifying independence assumption, what is the probability that at least one split is this discrepant? Why is the calculation only illustrative?

Solution

The family-wise probability is

$$p_{\rm glob}=1-(1-0.0455)^4\simeq0.170.$$

Thus selecting the largest of four turns a seemingly unusual local result into an event occurring about seventeen percent of the time under the null. The formula assumes independent tests and a search plan containing exactly those four splits. Real redshift, sky, scale, and catalog splits share objects and modes, and analysts may inspect additional diagnostics. Their global distribution should therefore be obtained from the complete analysis pipeline applied to null simulations.

Scope and open questions

The interpretation could change after a superseding CMB, BAO, supernova, or lensing release changes a discrepancy; after an independent calibration reveals a shared systematic; or after a model makes and passes a new prediction that materially resolves or strengthens the disagreement. A correction to covariance or selection is equally capable of changing the verdict.

The page supplies a robustness method and a dated evidence matrix. It does not publish undated best-fit values, declare that one current tension proves new physics, or guarantee that a disappearing marginal discrepancy validates the full baseline model.

References

• Abdul Karim, M., et al. (DESI Collaboration). 2025. “DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints.” Physical Review D 112: 083515. arXiv:2503.14738v3. DOI. Open PDF.
• Cortês, Marina, Ofer Lahav, and Andrew R. Liddle. 2026. “Cosmological Parameters.” In Review of Particle Physics, revised October 2025. Particle Data Group. Official PDF.
• Louis, Thibaut, et al. (ACT Collaboration). 2025. “The Atacama Cosmology Telescope: DR6 Power Spectra, Likelihoods and ΛCDM Parameters.” arXiv:2503.14452v2. Open PDF. Official DR6.02 release.
• Planck Collaboration. 2020. “Planck 2018 Results. VI. Cosmological Parameters.” Astronomy & Astrophysics 641: A6. DOI. Open PDF.