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.
- Redshift
The fractional change between observed and emitted wavelength after identifying the same spectral feature. In this module it is used to answer
What does a measured redshift actually tell us?
Observable/model connection: The term links
Spectroscopic line identification supplies wavelength ratios; the Pantheon+ excerpt provides heliocentric/CMB-frame redshift fields for supernovae
to Scale factor and the quantitative relation \(1+z = a(t_0)/a(t_e)\). State which part is measured, calibrated, derived, or model-dependent.Limit: At low z, peculiar velocities can be a substantial fraction of cz. Redshift alone is not a distance measurement. That boundary must remain attached when the term is used in a cosmological claim.
- Scale factor
The time-dependent relative size assigned to spatial separations between comoving observers in an FLRW model. In this module it is used to answer
What does a measured redshift actually tell us?
Observable/model connection: The term links
Spectroscopic line identification supplies wavelength ratios; the Pantheon+ excerpt provides heliocentric/CMB-frame redshift fields for supernovae
to Cosmological redshift and the quantitative relation \(1+z = a(t_0)/a(t_e)\). State which part is measured, calibrated, derived, or model-dependent.Limit: At low z, peculiar velocities can be a substantial fraction of cz. Redshift alone is not a distance measurement. That boundary must remain attached when the term is used in a cosmological claim.
- Cosmological redshift
Wavelength stretching associated with the change in scale factor between emission and observation. In this module it is used to answer
What does a measured redshift actually tell us?
Observable/model connection: The term links
Spectroscopic line identification supplies wavelength ratios; the Pantheon+ excerpt provides heliocentric/CMB-frame redshift fields for supernovae
to Peculiar velocity and the quantitative relation \(1+z = a(t_0)/a(t_e)\). State which part is measured, calibrated, derived, or model-dependent.Limit: At low z, peculiar velocities can be a substantial fraction of cz. Redshift alone is not a distance measurement. That boundary must remain attached when the term is used in a cosmological claim.
- Peculiar velocity
Motion relative to the local comoving expansion flow. In this module it is used to answer
What does a measured redshift actually tell us?
Observable/model connection: The term links
Spectroscopic line identification supplies wavelength ratios; the Pantheon+ excerpt provides heliocentric/CMB-frame redshift fields for supernovae
to CMB frame and the quantitative relation \(1+z = a(t_0)/a(t_e)\). State which part is measured, calibrated, derived, or model-dependent.Limit: At low z, peculiar velocities can be a substantial fraction of cz. Redshift alone is not a distance measurement. That boundary must remain attached when the term is used in a cosmological claim.
- CMB frame
A redshift reference frame corrected for the observer's motion relative to the microwave-background dipole. In this module it is used to answer
What does a measured redshift actually tell us?
Observable/model connection: The term links
Spectroscopic line identification supplies wavelength ratios; the Pantheon+ excerpt provides heliocentric/CMB-frame redshift fields for supernovae
to Redshift and the quantitative relation \(1+z = a(t_0)/a(t_e)\). State which part is measured, calibrated, derived, or model-dependent.Limit: At low z, peculiar velocities can be a substantial fraction of cz. Redshift alone is not a distance measurement. That boundary must remain attached when the term is used in a cosmological claim.
Physical and observational argument
A spectrograph measures wavelength, so redshift begins with line identification and wavelength calibration. For a comoving source and observer in an expanding FLRW spacetime, photon wavelengths scale with the cosmic scale factor during propagation, giving 1+z=a0/ae. Peculiar motion adds a local Doppler component; at low redshift it can dominate the Hubble-flow contribution. The Hubble parameter defines an instantaneous expansion rate, and c/H0 is a characteristic present scale, not the particle horizon or a direct distance to every object.
Cosmological redshift is a ratio of scale factors, not a Doppler velocity except in the low-redshift limit. Distinguish cosmological redshift from peculiar velocity and gravitational shifts before interpreting a spectrum.
Spectroscopic line identification supplies wavelength ratios; the Pantheon+ excerpt provides heliocentric/CMB-frame redshift fields for supernovae.
Derivation and interpretation
\[1+z = a(t_0)/a(t_e)\]
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.
At low z, peculiar velocities can be a substantial fraction of cz. Redshift alone is not a distance measurement.
Worked numerical anchor
At z=0.1, cz is 29,979.2 km/s as a low-z scale; the expansion interpretation requires the caveat in the lecture.
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
At low z, peculiar velocities can be a substantial fraction of cz. Redshift alone is not a distance measurement.
Spectroscopic line identification supplies wavelength ratios; the Pantheon+ excerpt provides heliocentric/CMB-frame redshift fields for supernovae.
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
- Convert between observed wavelength, emitted wavelength, and redshift with signs stated.
- Relate scale factor and redshift under metric expansion.
- Explain when cz is and is not an adequate recession-speed approximation.
Checkpoint: A galaxy has z=0.02. List two reasons that cz need not equal its total physical recession speed.
Answer in words before calculating; then identify one observation or null test that could falsify your interpretation.