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.
- FLRW metric
The homogeneous and isotropic spacetime geometry used to relate expansion history to photon paths. In this module it is used to answer
Why do cosmologists use several different distances?
Observable/model connection: The term links
The computation is independently checked against Astropy FlatLambdaCDM with radiation disabled to match the explicitly matter-plus-Lambda integral
to Comoving distance and the quantitative relation \(D_C(z)=c/H_0 integral_0^z dz'/E(z')\). State which part is measured, calibrated, derived, or model-dependent.Limit: Curvature changes transverse distance; radiation matters at early times. Never apply the flat, late-time integral to recombination as if it were a precision CMB model. That boundary must remain attached when the term is used in a cosmological claim.
- Comoving distance
The present-epoch radial coordinate separation obtained by integrating inverse expansion rate. In this module it is used to answer
Why do cosmologists use several different distances?
Observable/model connection: The term links
The computation is independently checked against Astropy FlatLambdaCDM with radiation disabled to match the explicitly matter-plus-Lambda integral
to Luminosity distance and the quantitative relation \(D_C(z)=c/H_0 integral_0^z dz'/E(z')\). State which part is measured, calibrated, derived, or model-dependent.Limit: Curvature changes transverse distance; radiation matters at early times. Never apply the flat, late-time integral to recombination as if it were a precision CMB model. That boundary must remain attached when the term is used in a cosmological claim.
- Luminosity distance
The distance definition that makes bolometric flux obey the inverse-square relation. In this module it is used to answer
Why do cosmologists use several different distances?
Observable/model connection: The term links
The computation is independently checked against Astropy FlatLambdaCDM with radiation disabled to match the explicitly matter-plus-Lambda integral
to Angular-diameter distance and the quantitative relation \(D_C(z)=c/H_0 integral_0^z dz'/E(z')\). State which part is measured, calibrated, derived, or model-dependent.Limit: Curvature changes transverse distance; radiation matters at early times. Never apply the flat, late-time integral to recombination as if it were a precision CMB model. That boundary must remain attached when the term is used in a cosmological claim.
- Angular-diameter distance
Physical transverse size divided by observed angular size. In this module it is used to answer
Why do cosmologists use several different distances?
Observable/model connection: The term links
The computation is independently checked against Astropy FlatLambdaCDM with radiation disabled to match the explicitly matter-plus-Lambda integral
to Distance duality and the quantitative relation \(D_C(z)=c/H_0 integral_0^z dz'/E(z')\). State which part is measured, calibrated, derived, or model-dependent.Limit: Curvature changes transverse distance; radiation matters at early times. Never apply the flat, late-time integral to recombination as if it were a precision CMB model. That boundary must remain attached when the term is used in a cosmological claim.
- Distance duality
The relation D_L=(1+z)^2 D_A for transparent metric propagation with photon conservation. In this module it is used to answer
Why do cosmologists use several different distances?
Observable/model connection: The term links
The computation is independently checked against Astropy FlatLambdaCDM with radiation disabled to match the explicitly matter-plus-Lambda integral
to FLRW metric and the quantitative relation \(D_C(z)=c/H_0 integral_0^z dz'/E(z')\). State which part is measured, calibrated, derived, or model-dependent.Limit: Curvature changes transverse distance; radiation matters at early times. Never apply the flat, late-time integral to recombination as if it were a precision CMB model. That boundary must remain attached when the term is used in a cosmological claim.
Physical and observational argument
For radial light propagation, ds=0 in the FLRW metric gives a comoving interval equal to the integral of c dt/a(t), or c dz/H(z). Curvature maps radial comoving distance into a different transverse distance; only in a flat geometry are they equal. Flux dilution and photon-energy/time stretching lead to luminosity distance, while apparent angular size uses angular-diameter distance. Etherington reciprocity links them when photon number is conserved and propagation is metric. Every distance label therefore encodes a measurement operation and assumptions.
The homogeneous-isotropic FLRW metric defines line-of-sight comoving, transverse comoving, angular-diameter, luminosity, and volume-averaged distances. They answer different measurement questions and are linked by reciprocity, not interchangeable by name.
The computation is independently checked against Astropy FlatLambdaCDM with radiation disabled to match the explicitly matter-plus-Lambda integral.
Derivation and interpretation
\[D_C(z)=c/H_0 integral_0^z dz'/E(z')\]
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.
Curvature changes transverse distance; radiation matters at early times. Never apply the flat, late-time integral to recombination as if it were a precision CMB model.
Worked numerical anchor
At z=1, D_C=3401.263 Mpc, D_L=6802.526 Mpc, and D_A=1700.631 Mpc.
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
Curvature changes transverse distance; radiation matters at early times. Never apply the flat, late-time integral to recombination as if it were a precision CMB model.
The computation is independently checked against Astropy FlatLambdaCDM with radiation disabled to match the explicitly matter-plus-Lambda integral.
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
- Derive the radial comoving-distance integral from the FLRW line element.
- Use distance duality D_L=(1+z)^2 D_A under transparent metric propagation.
- Compute and label D_C, D_L, D_A, and D_V with units.
Checkpoint: Check that c/H0 is a length and that every reported distance is in Mpc.
Answer in words before calculating; then identify one observation or null test that could falsify your interpretation.