Preparation: algebra, introductory mechanics, and the previous module's stated prerequisites. This is a 60-minute instructional module; use the slide timings as a guide and pause at the active check.
Terms Developed in Context
These terms organize the chain from recorded signal to calibrated observable, likelihood, and bounded cosmological claim.
- Concordance
Consistency of multiple probes within a shared model and correctly represented covariance. In this module it is used to answer
When do two experiments disagree?
Observable/model connection: The term links
All comparison points are literature summaries and are tagged to their exact analysis assumptions in the dataset table and reference log
to Tension and the quantitative relation \(Delta theta = theta_A - theta_B; significance requires Cov(theta_A,theta_B)\). State which part is measured, calibrated, derived, or model-dependent.Limit: Do not calculate a 'sigma tension' by adding published errors in quadrature unless independence and comparable posterior shapes are defensible. That boundary must remain attached when the term is used in a cosmological claim.
- Tension
A quantified discrepancy between inferences that accounts for uncertainty, dependence, and parameterization. In this module it is used to answer
When do two experiments disagree?
Observable/model connection: The term links
All comparison points are literature summaries and are tagged to their exact analysis assumptions in the dataset table and reference log
to Shared covariance and the quantitative relation \(Delta theta = theta_A - theta_B; significance requires Cov(theta_A,theta_B)\). State which part is measured, calibrated, derived, or model-dependent.Limit: Do not calculate a 'sigma tension' by adding published errors in quadrature unless independence and comparable posterior shapes are defensible. That boundary must remain attached when the term is used in a cosmological claim.
- Shared covariance
Statistical dependence between probe estimates caused by common data, calibration, volume, or modeling. In this module it is used to answer
When do two experiments disagree?
Observable/model connection: The term links
All comparison points are literature summaries and are tagged to their exact analysis assumptions in the dataset table and reference log
to Posterior predictive check and the quantitative relation \(Delta theta = theta_A - theta_B; significance requires Cov(theta_A,theta_B)\). State which part is measured, calibrated, derived, or model-dependent.Limit: Do not calculate a 'sigma tension' by adding published errors in quadrature unless independence and comparable posterior shapes are defensible. That boundary must remain attached when the term is used in a cosmological claim.
- Posterior predictive check
A comparison between observed diagnostics and data replicated from the fitted model. In this module it is used to answer
When do two experiments disagree?
Observable/model connection: The term links
All comparison points are literature summaries and are tagged to their exact analysis assumptions in the dataset table and reference log
to Look-elsewhere effect and the quantitative relation \(Delta theta = theta_A - theta_B; significance requires Cov(theta_A,theta_B)\). State which part is measured, calibrated, derived, or model-dependent.Limit: Do not calculate a 'sigma tension' by adding published errors in quadrature unless independence and comparable posterior shapes are defensible. That boundary must remain attached when the term is used in a cosmological claim.
- Look-elsewhere effect
Inflation of apparent significance when many discrepancies or subsets are searched and only the largest is reported. In this module it is used to answer
When do two experiments disagree?
Observable/model connection: The term links
All comparison points are literature summaries and are tagged to their exact analysis assumptions in the dataset table and reference log
to Concordance and the quantitative relation \(Delta theta = theta_A - theta_B; significance requires Cov(theta_A,theta_B)\). State which part is measured, calibrated, derived, or model-dependent.Limit: Do not calculate a 'sigma tension' by adding published errors in quadrature unless independence and comparable posterior shapes are defensible. That boundary must remain attached when the term is used in a cosmological claim.
Physical and observational argument
Two experiments disagree only after they estimate the same parameter under compatible assumptions. For a difference, the variance includes both individual variances and their covariance; shared calibration or common external data can make that covariance nonzero. Posterior shapes may also be non-Gaussian or prior-sensitive. A responsible comparison first harmonizes definitions, data cuts, nuisance models and parameterization, then evaluates a difference using joint samples or a justified covariance model. The course summary table deliberately preserves source labels and does not manufacture a combined tension statistic from incomplete marginal errors.
A tension claim requires harmonized parameter definitions, assumptions, and uncertainty models. Shared data, calibrations, or theory priors create covariance; scalar error-bar separation is only a diagnostic under strong independence assumptions.
All comparison points are literature summaries and are tagged to their exact analysis assumptions in the dataset table and reference log.
Derivation and interpretation
\[Delta theta = theta_A - theta_B; significance requires Cov(theta_A,theta_B)\]
Begin by naming observables, coordinate convention, and model assumptions; each algebraic transformation must preserve units and the distinction between measured and inferred quantities. Test a limiting case, then perturb one input and state which observable changes.
Do not calculate a 'sigma tension' by adding published errors in quadrature unless independence and comparable posterior shapes are defensible.
Worked numerical anchor
Probe summaries in the package are not combined because the underlying likelihoods and cross-probe covariance are not included.
Reproduce the value with the course computation module, retain full precision in code, and round only for communication. Report units and all conditional assumptions with the result.
Evidence, systematics, and limits
Do not calculate a 'sigma tension' by adding published errors in quadrature unless independence and comparable posterior shapes are defensible.
All comparison points are literature summaries and are tagged to their exact analysis assumptions in the dataset table and reference log.
Separate the directly recorded observable, its calibration, the derived intermediate quantity, and the model-dependent parameter. An uncertainty bar is not a complete likelihood when shared covariance or selection is omitted.
Study and retrieval practice
- Define a parameter-level comparison with common conventions.
- Identify shared-data and shared-calibration covariance pathways.
- Write a defensible checklist before assigning a numerical tension significance.
Checkpoint: Specify the metadata required before a joint analysis can be run.
Answer in words before calculating; then identify one observation or null test that could falsify your interpretation.