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.
- Selection function
The probability that an object with specified latent and observed properties enters the analyzed sample. In this module it is used to answer
Which objects never enter the catalog?
Observable/model connection: The term links
The release documentation and published sample counts are cited; no survey completeness map or detection efficiency curve is bundled
to Malmquist bias and the quantitative relation \(p(detected|x) = S(x) p(x) / integral S(x')p(x') dx'\). State which part is measured, calibrated, derived, or model-dependent.Limit: An observed histogram cannot by itself identify completeness because it mixes cosmological volume, population evolution, and selection. That boundary must remain attached when the term is used in a cosmological claim.
- Malmquist bias
Preferential inclusion of intrinsically brighter objects in a flux-limited sample. In this module it is used to answer
Which objects never enter the catalog?
Observable/model connection: The term links
The release documentation and published sample counts are cited; no survey completeness map or detection efficiency curve is bundled
to Eddington bias and the quantitative relation \(p(detected|x) = S(x) p(x) / integral S(x')p(x') dx'\). State which part is measured, calibrated, derived, or model-dependent.Limit: An observed histogram cannot by itself identify completeness because it mixes cosmological volume, population evolution, and selection. That boundary must remain attached when the term is used in a cosmological claim.
- Eddington bias
Net migration across a threshold caused by measurement scatter and a steep underlying population distribution. In this module it is used to answer
Which objects never enter the catalog?
Observable/model connection: The term links
The release documentation and published sample counts are cited; no survey completeness map or detection efficiency curve is bundled
to Completeness and the quantitative relation \(p(detected|x) = S(x) p(x) / integral S(x')p(x') dx'\). State which part is measured, calibrated, derived, or model-dependent.Limit: An observed histogram cannot by itself identify completeness because it mixes cosmological volume, population evolution, and selection. That boundary must remain attached when the term is used in a cosmological claim.
- Completeness
The fraction of the target population retained by detection and analysis cuts. In this module it is used to answer
Which objects never enter the catalog?
Observable/model connection: The term links
The release documentation and published sample counts are cited; no survey completeness map or detection efficiency curve is bundled
to Injection-recovery and the quantitative relation \(p(detected|x) = S(x) p(x) / integral S(x')p(x') dx'\). State which part is measured, calibrated, derived, or model-dependent.Limit: An observed histogram cannot by itself identify completeness because it mixes cosmological volume, population evolution, and selection. That boundary must remain attached when the term is used in a cosmological claim.
- Injection-recovery
A calibration procedure that inserts controlled synthetic sources and measures their probability of recovery. In this module it is used to answer
Which objects never enter the catalog?
Observable/model connection: The term links
The release documentation and published sample counts are cited; no survey completeness map or detection efficiency curve is bundled
to Selection function and the quantitative relation \(p(detected|x) = S(x) p(x) / integral S(x')p(x') dx'\). State which part is measured, calibrated, derived, or model-dependent.Limit: An observed histogram cannot by itself identify completeness because it mixes cosmological volume, population evolution, and selection. That boundary must remain attached when the term is used in a cosmological claim.
Physical and observational argument
A catalog is a selected sample, not a transparent window onto the parent population. Detection probability can depend on flux, color, cadence, host background and follow-up resources. Near a flux threshold, more numerous faint sources can scatter upward than bright sources scatter downward, producing Eddington bias; Malmquist bias describes related luminosity-selection effects. The likelihood must account for truncation and selection normalization. A redshift histogram mixes comoving volume, evolution and completeness, so the deliberately target-stratified Pantheon+ excerpt is useful for plotting practice but cannot estimate the survey selection function.
A flux-limited survey samples a population through a selection function. Detection thresholds, cadence, color cuts, spectroscopic follow-up, and analysis cuts shape the observed redshift and luminosity distributions.
The release documentation and published sample counts are cited; no survey completeness map or detection efficiency curve is bundled.
Derivation and interpretation
\[p(detected|x) = S(x) p(x) / integral S(x')p(x') dx'\]
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.
An observed histogram cannot by itself identify completeness because it mixes cosmological volume, population evolution, and selection.
Worked numerical anchor
The 18 Pantheon+ redshifts were selected near fixed targets, so their histogram cannot estimate the parent survey completeness.
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
An observed histogram cannot by itself identify completeness because it mixes cosmological volume, population evolution, and selection.
The release documentation and published sample counts are cited; no survey completeness map or detection efficiency curve is bundled.
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 selection function and distinguish it from an observed distribution.
- Explain Malmquist and Eddington biases with the direction of the effect stated.
- Design a completeness diagnostic that uses injection/recovery or external calibration.
Checkpoint: Explain forward modeling of selection and why truncating a likelihood after fitting is generally wrong.
Answer in words before calculating; then identify one observation or null test that could falsify your interpretation.